Title

Probabilistic Aspects of Besicovitch Almost
Periodic Functions

Author(s)Trinh, Khanh Duy

Citation

大阪大学, 2012, 博士論文

Version Type VoR
URL

https://hdl.handle.net/11094/27473

rights
Note

Osaka University Knowledge Archive : OUKA
https://ir.library.osaka-u.ac.jp/
Osaka University

1'4ι

Probabilistic Aspects of Besicovitch
AImost Periodic FuncttonS

Trinh Khanh Duy

Department of Mathematics,
Graduate School of Science, Osaka University

April 20Lz

シ
靖お
でノ

Probabilistic Aspects of Besicovitch
Almost Periodic Functions(ベ シ コ ビ ッチ 概 周 期 関数 の確 率 的側 面 )Trinh Khanh Duv

Department of Mathematics,
Graduate School of Science, Osaka University

April 2012

Preface

ゥ
ヽ

進

ご

∫

′ ヽ

＼

ノ9:=
‖

・一
π

／

中
﹄
・

The theory of almost periodic functions, established by H. Bohr, fl&y be considered fromtwo different points of view. On the one hand, an almost periodic function is a continuousfunction possessing a certain structural property which is a generalization of pure periodic
functions, and on the other it is the limit of trigonometric polynomials under the uniform
norm.
Corresponding to the two different points of view, further development of the theoryof almost periodic functions to generalize this theory went in two different directions. The
first direction was developed by W. Stepanov and H. Weyl, which led to two importantclasses of almost periodic functions: Stepanov almost periodic functions and Weyl almost
periodic functions.
The second direction of generalizations was that followed by A.S. Besicovitch. Besicovitch enlarged the class of almost periodic functions by considering the convergence ofsequences with respect to some Besicovitch q-(semi)norm (1 < S < oo) rather than uniform
convergence. Here the Besicovitch q-norm of a function /: R. -+ C is defined as

Then a function is called Besicovitch almost periodic function if it is a limit of trigonometricpolynomials under some Besicovitch q-norm. Besicovitch almost periodic functions look
Iike random variables because they possess mean values and limit distributions. On theother hand, iike periodic functions, they have "Fourier series" in the form of a general
trigonometric series,f (f )- Lo^r,^*,

.

ヽ

／
ヽ

︑ ︲ ′ ／

η

ノ

′ ︲ ｌ

-A/-+oo

１一Ⅳll/lls :: Iimsup

Ⅳい
″祠
︐

For arithmetical functions (functions defined on N), a function /: N -+ C is called
Besicovitch almost periodic arithmetical function if it belongs to the linear closure of{eo: a e RIZ,} under some Besicovitch g-(semi)norm (1 ( q < oo)where eo stands for the function e(t: n +. Besicovitch almost periodic arithmetical"2rianfunctions also have mean values, limit distributions and "Fourier series".
This research is to study particular classes of Besicovitch functions: Besicovitch almost
periodic functions with Fourier exponents belonging to a Dirichlet sequence and Besicov-

itch limit-periodic arithmetical functions. We will present recent results on the following
problems.

(1) Constructing probability spaces where Besicovitch functions in these classes can beconsidered as random variables.

PrefacelV(2) Convergence of Fourier series.
(3) Some applications: value distributions of general Dirichlet series; natural extensionsof additive/multiplicative arithmetical functions, and the distribution of k-th power
free integers.

This thesis is mainly taken from the author's five papers ([10, 1I,1,2,13, 14]), and isdivided into two parts corresponding to two classes of Besicovitch functions.
Part I deals with Besicovitch almost periodic functions, that is, functions /: IR. -+ C
which belong to the closure of trigonometric polynomials under some Besicovitch q-norm(1 ( q < m). In this case, the mean value of a function / is defined as

1 f'r

Ml/l : ,I l-o J_rt1)at.

provided that the limit exists, and the limit distribution is the probability distribution on
C to which the sequence of probability measuresu7{r : f(r) e A}, A e B(C.),
converges as ? -+ oo, where z7 denotes the uniform probability measure on [-?,?] .

For every Besicovitch almost periodic function f , the mean valuea()) : Mlf (t)e-i^'l' : .lim : f -f p)e-i^tdt?-roo Zi J _f
exists for all .\ e JR., and those ) for which a()) is non-zero are at most countable, called
the Fourier exponents of /. The formal seriesf (t) - !c(,\)ei)t,\

is called the Fourier series of /.
In Chapter 1, we study Besicovitch almost periodic functions with Fourier series of the

forms

m

f(t)-lo^r-'^-'.
rn:l

where {,\-} is a strictly increasing sequence of non-negative numbers tending to infinity,
called a Dirichlet sequence. We will construct a suitable probability space where thesefunctions f can be extended to random variables. For these functions, their Fourier series
are shown to be convergent in norm with the usual order (1 < q < m). This resrrlt issimilar to the convergence in norm of classical Fourier series. Besides, a version of theCarleson-Hunt theorem is investigated.
Chapter 2 concerns with general Dirichlet series of the formS-\)_,a^e-n*t. s:ollf
where a^ € C,and€C.

{)-} ,, rl-rrr"ntet sequence. Suppose that the above series converges

absolutely for o ) oo and has the sum /(s). Then /(s) is an analytic function in thehalf-plane D :: {s€ C : o ;, oo}.

Preface

Assume that the function /(s) is meromorphically continuable to a wider half-plane
Ds :: {s e C : o } o6),oo I oaand satisfies some mild conditions. Then for fixed o } os,f (o + zf) is a Besicovitch almost periodic function with the Fourier seriesf (o +if ) - ia^s_ ;^or-i\'nt.

m:7

Thus the limit distribution of f (o * if) is well identified by using the probability spacedeveloped in Chapter 1. Moreover, using this probability space, we can also identify limit
distributions of general Dirichlet series in the space of analytic functions and in the spaceof meromorphic functions.
Part II deals with Besicovitch limit-periodic arithmetical functions, that is, functions/: N -+ C which belong to the closure of periodic functions under some Besicovitch qnorm. The mean value of a function f is defined as.

,A'I

Ml/l :: ;*#|fl"1.
n:I

and the limit distribution is considered as follows: if the limit-I/hm(zsRe f (") +i,trmf (n,)), (",t) e tR2,
N--+oo |rV )_-""0fexists and it coincides with the characteristic function of some probability distribution
on lR2 3 C, then we call it the limit distribution of /. Note that the space of periodic
arithmetical functions is spanned by {"olr:r:I,2,...,1 ( cr,( r,gcd(a,r) :1}. Thus, alimit-periodic arithmetical function / is a Besicovitch almost periodic arithmetical function
with the following Fourier series∫(η )〜calreolr\n )t,.,Jf,).,,:,
where calr t: M[/(" Wnl

and (a,r) denotes the greatest common divisor gcd(c,r) of

a and r.
Let Dq denote the space of q-limit-periodic arithmetical functions and let Dq be thequotient space of Oq with respect to the null-space Af (p01 ,: { f € Dq : ll/lln : 0 }. Let
2 b" th" ring of finite iltegral adeles with its normalized Haar measure ).
In Chapter 3, (Z,B(Z),.\) is shown to be a good probability space where limit-periodic
arithmetical functions can be considered as random variables. In fact, every function in
Dq can be extended to a random variable in Lq (2,1). The limit distribution of the originalfunction coincides with the distribution of the extended random variable. In addition, thespace Da is isometrically isomorphic to Ls(Z,\.
Dealing with the problem of convergence of Fourier series, for each n € N, we define a
finite Fourier expansion of a function f e Dq as
S,t(∫ ):=(∫ ,Cα /r)Cα /r.

r22;
1く αくr;(α ,r)=1

Then our result is that,for lく 9<OO,
Sれ (ノ )一 ノ
￨19→

O as η→ Oin Z, ノ∈D9.

Preface

Vl

This gives an approximation for limit-periodic arithmetical functions by periodic arithmetical functions. The natural extensions of additive and multiplicative arithmetical functions will be considered at the end of this chapter.
Chapter 4 deals with the distribution of k-th power free integers. Let yw) (n) be theindicator function of the set of k-th power free integers, that is,l, (Vp : prime pk n),v@) t^,
1L\"i ._'- I o, (3p : prime,, pk Il r).
\
Then X(k)(n) is a multiplicative limit-periodic arithmetical function. The mean value ofY{r,) @) exists and it is equal torilI

Iim *fX,ol(n)=-MfX{t)1-L-- , : N;;((,k),
N 2_
where ( is the Riemann zeta function. Using a result in Chapter 3, we exten4 ;(r) (n) to a
in a natural way. Then investigating
random variable on the probability space (2, B(Z),^)

the rate of .L2-converqence ofc(k)1.1 ._t:5),''(r).Nr \-rz(klr - , .r

N \Xt^'(r*n).

r €2.

we obtain the following result

1 M /

/

1 rr2

;'+np, (t (tif',-,- #)) * cortst Nt/r
Note that a conjecture thatvE >: o (N,/2t+e)/ . A/ -+ oo.
0. x (sf,)t*) - l)
((k)/
\
\-u'

has not been proved yet. Our result may be called as a mean square version of thisconiecture.

Vll

Acknowledgments
First and foremost, I would like to express my sincere gratitude to my supervisor, Professor
Hiroshi Sugita, for his great encouragement and insightful guidance during my graduate
study at Osaka University. Not only has he given me excellent research advice and patiently corrected my writing, but he also has shared many interesting experiences about
Japanese culture. In addition, without his great help, I would have never been able tocomplete any application forms in Japanese. I cannot say thank enough for his tremendous
support and concern.
I also sincerely would like to thank Professor Satoshi Takanobu, Professor Shinichi
Kotani, and Professor Kohji Matsumoto for giving me valuable advice during my work.
Besides, I wish to express my gratitude to the faculty of Department of Mathematics
and the fellow graduate students for their consideration and cooperation in this effort.
Finally, I wish to give my special thanks to my family and my dear friends who werealways thoroughly understanding me and heartily encouraging me during my stay in Japan.
This work is sponsored by Matsuda Yosahichi Merorial Foreign Student Scholarship;
the Ministry of Education, Culture, Sport, Science and Technology of Japan (MEXT);and the Research Fellowships of the Japan Society for the Promotion of Science for Young
Scientists (JSPS Research Fellowships for Young Scientists).
Osaka, April 2012

Trinh Khanh Duy

Contents
PrefaceAcknowledgments
Notations and symbols .

iii
vii
xi

O Preliminaries
0.1 Convergence of probability measures
0.1.1 Weak convergence in metric spaces
0.7.2 Convergence of random elements

1
1
1

2

.....
0.1.3 Weak convergence in lRd and characteristic functions
0.1.4 Weak convergence in topological groups and Fourier transforms

0.2 Ergodic theory
0.2.7 Discrete time

3

4
4

.....
.....

0.2.2 Continuoustime

0.3 Martingales

0.3.1 Conditional expectation
0.3.2 Martingales, definition and convergence theorems

I

2

Besicovitch Almost Periodic F\rnctions

6
8
8

I
13

Besicovitch Almost Periodic Functions With Fourier Exponents Belonging to a Dirichlet Sequence15
1.1 Introduction.
15

L.2 General theory of Besicovitch almost periodic functions
1.3 Probabilitv space associated with Besicovitch almost periodic functions
1.3.1 Besicovitch functions whose Fourier exponents belong to a subgroup
7.3.2 Besicovitch functions whose Fourier exponents belong to a Dirichlet

16
17
77
27

seouence23
1.4 Convergence results.....
23
I.4.1. ClassicalFourierseries
periodicfunctions25
1.4.2 Convergence of Fourier series of Besicovitch almost
1.4.3 The linearly independent case30

2 Value Distributions of General Dirichlet Series
2.7 Introductiorr
2.2 General theory
2.3 Limit theorems for general Dirichlet series on the complex plane
2.4 Limit theorems for eeneral Dirichlet series in functional sDaces
2.4.l

Absolutely convergent case

33
33
35
39
40
40

2.4.2 General case

4I
ix

Contents

II

Besicovitch Limit-periodic Arithmetical Funtions45

3 Limit-periodic Arithmetical Functions and The Ring of Finite Integral
Adeles47

3.1 Introduction.
3.2 The ring of finite integral adeles

47
49
49

3.2.7 The ring of finite integral adeles and some basic properties
3.2.2 Connection to limit-periodic arithmetical functions

Index

５
８

Bibliography

つ４ ″
７
７

,S,r,,

１
´
７

.

■
一
７

4 The Distribution of k-th Power Flee Integers
4.L Introductiorr
4.2 Explicit formula for
4.3 Estimate of the -L2-norm and limit points in ,L2
4.4 Mean square convergence rate¨
５
６

3.4 Additive and multiplicative arithmetical functions︵
６
６

3.3.1 Convergence of Fourier expansions in Lq(Z,A) . . .
3.3.2 Convergence of Fourier expansions of limit-periodic functions
3.3.3 Convergence of Fourier expansions of almost-even functions

53
60

∩Ｕ ︵Ｚ
６
６

3.3 Convergence of Fourier expansions

Notations and symbols

Xl

Notations and symbols
A:: B
A is defined bv B (B:: A as well).
weak convergence of probability measures.
+*))convergence in distribution of random elements.
Ls-:-+
convergence in Lq.
b is divisible by a.
"lb
L.H.S.
left-hand side.
right-hand side.
R.H.S.
i.i.d.
independent and identically distributed.
w.r.t.
with respect to.
a.e.
almost everywhere.
a.s.
almost surely.
N :: {I,2,.. . }, the set of all natural numbers.
Z :: {. . . , -2, -1,0, 7,2, . .. }, the set of all integers.
A :: the set of all rational numbers.
R :: the set of all real numbers.
C :: the set of all complex numbers.
gcd(a, b) :: the greatest common divisor of a and b.
lcm(a, b) :: the least common multiple of a and b.
J4 or i :: the imaginary unit.
B(S) :: the Borel (o-)field of a topological space ,S.
1 rrlF

Mf/l ::
E[X] i:B{e) [X]::llf llr"6,n',,4 ::

1jl

_lim
I f ftlat. or _lim
/ -f (t)dt.or 1v-*
^lim *)
T--+x .ll J_7T-+x: |I Jto
1\.f t"),-_1'

the mean value of function /.
the mean (expectation) of random variable X, (the probability
space is clear in the context).
the mean (expectation) of random variable X with respect to theprobability measure P.
/ f\1/q(llf ll"" or llf llr'6,,"7 as well), wherelf@)lqdm(r)
I

I
\Jx

/I

(X,B,rn) is a measure space.

Chapter 0

Preliminaries
0.1 Convergence of probability measures
0.1.1 Weak convergence in metric spaces
Let(S,p) beametricspaceandlet6(s) denotetheBorelo-field of ^9. Let{P2},,65and
P be probability measures on (S,6(5))'
Definition 0.1-. We say that {Pr,} converges weakly to P as n -+ 6i and write P,, + P,
if for all bounded continuous functions ./: 5 + R,lim

I for,: I for.

n_+(e J q

JS

Since two probability measures P and Q coincide it Isf dP: fsf dq for all bounded,uniformly continuous real functions /, it follows that the sequence {Pr} cannot converge
weakly to two different limits.
The Portmanteau theorem provides useful conditions equivalent to weak convergence.

Theorem O.2. The followi'ng fiue conditions are equ'iualent:(i) P" ? P;(ii) "G f d,Pn -+ I, f ap for all bounded, uni'formly cont'inuous f ;(iii) limsup,, P,"(F) < P(F) for all closed F;(iv) Iim inf P^(G) >- P (G) for all open G ;"(v) Pr(A) -+ P(,a) for allP-conttnur,ty sets A. Here aP-cont'inui,ty set'is a set A whoseboundary 0A satisfi'es P(0,4) :O.

This is Theorem 2.1 from Billingsley [4]'Let (,9,, p,) be another metric space and let 6(5') be the Borel o-field of S'. We
consider a measurable (or B(S) 16(,5')-measurable) mapping h: S -+ ,S', that is, a mapping
h satisfies

h-t(9,s'D c 6(s).
Then each probability measure P on (^9, 6(5)) induces on (,5" B(S')) a probability measureph-1 defined by Ph-l(A) :P(h-|A),A e B(S'). Let Dl denote the set of discontinuities
of h. Then Dh € B(S) and we have the mappi'ng theorem.

Theorem o.3. If P,, * P and' P(Dil : 0, then P"h-: + Ph-1 ' In partr'cular' i'fh: S -+ St i,s conti,nuous and Pr, * P, thenPnh-t +Ph-'.
This is Theorem 2.7 from Billingsley [4].

Chapter 0. Preliminaries

1.2

0。

Convergence of randollrl elements

A mapping X fl・ om a probability space(Ω ,F,P)tO a metric space(S,ρ )iS Said tO be arandom element ifit is F/β (S)― measurable,that is,Xl(β (S))⊂ F・
An Rζ ―
valued or C― valued randolln element is usually called a random variable. The d,strj―

う包ιづθη of χ is the probability measure Px=PX l on(S,β1ス

Px(■ )=P(χ(S))deined by)=P(ω :χ (ω )∈ ス)=P(X∈ ■)・

Let{χ πN and X be random elements.
}η ∈石e stt that the sequence{Xη }COnVerges in distribution to χ,and write
Deinition O.4.ヽ ヽxη ム

χ,if the sequence of distributions(Pxη }COnVerges weakly to Px as η→ ∞.

If X and y are S― valued random elements deined on the same prObability space(Ω ,F,P),then it makes sense to speak of the distance ρ(X,y)・ In the sequel,let(S,ρ )be a separable metric space.Then ρ(X,y)iS a random ttriable.
Deinition O.5.ヽ e stt that the sequence{X72}COnК rgё s in probability to χ if for every
1石

ε>0,
鳳

P(ρ (χれ,X)≫ ε)=0・づ
Theorem O。 6. Tん c cθ ηυθη θηCC tη prο bα bづ ιη αづsι れbzι づθ2.
づCS tん c cθ 77υ θηpι づη θ26θ づ̀νThis is a consequence of Theorem 3.l from Billingsley[41.

Theorem O。 7.Zθ ι{Zら }η ∈N,{χ た}に N αηα{χ た,n}た,れθm cι θttθ η;s.■ sttθ J ttη α∈N ιC S― υαιsamc tん αι←)χ た,η ム

χた

αs

χ たム

χ

αsた → ∞

η → oo;(ii)

η ε>0,(iii)/Or eυ θ
鳳sη → ∞
動 cη 耽 ム χ α;εhm Sup P(ρ (χ た)=0・,れ ,L)≫.

This is TheOrem 3.2 fl・ om Billingsley 14].

0。

1.3

Weak convergence in Rd and characteristic functions

The characteristic fllnctionり に)of a probability metture P on(Rd,β (Rd))iS deined by

αJ=(dp咄
where(■ ,■ )denotes the inner product ofェ and″ in Rd.Note that the characteristic
function 9 uniquely deterlllines the probability measure]P.

0.1. Convergence of probability measures
Theorem O.8. Let {Pn} and,P be probabi,li,ty measures on (Rd,6(Rd)) and let gn(r) andgQ) be the correspondzng characteri,sttc functi,ons. Then Pr, * P, i'f and only i'fgn\r1 -+ p(r) for alt r e Rd.
The following theorem is L6vy's famous continuity theorem.

Theorem O.9. Let {P,} be probabi,li,ty rneasures an (1Rd,6(Rd)) and let p,(r) be thecorrespondi,ng characterzsti,c funct'ions. Assume thatpn|) -+ plr) for all r e Rd.
andthatgQ) i,s continuous atthepoi.ntr: (0,...,0). Thentherei,s aprobabi,l'ity measure
P on (Rd,6(Rd)) such, that P", * P, and 9Q) i,s the characteri'sti,c functi'on of P.
Proofs of these theorems can be found in Section 29 of Billingsley [4].

0.1.4 Weak convergence in topological groups and Fourier transforms
A measure P on (,S,6(5)), ,S being a compact topological space, is said to be regular iffor every e > 0 and every E e B(S), there is a compact set M and an open set I/ with
M c E C [/ and P(U\ M) < e. It is known that if 5 is metrizable, then any probability
measure on (S,6(5)) is regular. Let G be a compact topological group. A measure P on
(G,B(G)) is said to be invariant if
P(A) : P(rA):P(Ar)for all A e B(G) and all r €. G, where rA and Az denote the sets {ry t ll e A} and{Ar t y e A}, respectively.
Theorem O.1O. Let G be a compact topologi,cal grow. Then there'is a uni;que'inuariant
regular probabtlzty measure mH on (G, B(G)), called the normali.zed Haar rneasure.
Now let G be a locally compact abelian group. Let G denote the collection of allcontinuous homomorphisms of G into the unit circle 7 : {z €C : lzl : 1}. The membersof d are called the characters of G. Under the operation of pointwise multiplication offunctions, G i. un abelian group. With the compact open topology, G becomes a locallycompact abelian group.
We have the following results (see also Walters 146, Section 0.7] and its references).

(i) G is compact, if and only if d is discrete.

(ii) (Duality theorem). (6) ir naturally isomorphic (as a topological group) to G, theisomorphism being given by the mapping G > g r+ Xg, where XnQ): XG) for allX€G.

(iii) If Gt,Gz are locally compact abelian groups, then G1??" : dr x Gz. (Here"x" denotes the direct product.) Helce all characters of G1 x G2 are of the form(g,h)+xk)d(h), where X€Gr,6e Gz.
(iv) If f is a subgroup of d, thenH:{geG:y(g):1,VX€f}
is a closed subgroup of G and 1QE\ : r.

Chapter 0. Preliminaries(v) If fI is a closed subgroup of G and H + G,then there exists a character X € e, X + 7
such that X(h) : l- for all h e H.
(vi) Let G be compact. Then finite linear combinations of characters are dense inC(G),the space of complex-valued continuous functions on G. The members of d form an
orthonormal basis for L2 (G, m 11).
We get back to the case G being compact abelian group. The Fourier transform of aprobability measure P on (G, B(C)) is a function defined on the characters gronp d,g(r): I r(g)aP(g). re G.
JG

By the property (vi) above, it is clear that 9(1) uniquely determines the probability
measure P. Let 16 denote the trivial character of G, the character that is identicallyequal to 1. Then the Fourier transform of the normalized Haar measure mH is as follows

9(\) :t" ',Jr\\9)amu(9t

if 1:10'10. if r I ro.

: [t'We have the following continuity theorem.

Theorem O.IL. Let G be a compact abeh,an group. Let {P"} be probabi.li,ty measures on(G,B(G)) and g"(y) be the correspond'ing Fourier transforms. Assume thats,(x) -+ s(x) fo, aU v e G.
Then there i,s a probabi,li,ty measureP on (G,B(G)) such thatP, )P,
Fourier transform of P.

and g(y) is theThis is a special case of Theorem 1.4.2 from Heyer [23].

O.2 Ergodic theory
O.2.I Discrete time
This section is taken from Chapter 1 of Walters [46].
Let (X, B,m) be a probability space.

Definition 0.12. (i) A transformation T: X -+ X is said tobe measurable if :f-r lq CB.

' (ii) A transformation 7: X -+ X is said to be measure-preseruing if 7 is measurable
and m(T-rB) : m(B) for all B e B.

LetT: X -+ X be ameasure-preservingtransformation. If T-rB: B for B e B,thenalso ?-1(X\B) : X\B and we could study ? by studying the two simpler transformations(or
716 and Tlxts. If 0 < m(B) ( 1, this has simplified the study of T. If^(B):07 sincem(B): 1), we can ignore B (or X \ B) and we have not significantly simplified
neglecting a set of zero measure is allowed in measure theory. This raises the idea ofstudying those transformations that cannot be decomposed as above and of trying to
express every measure-preserving transformation in terms of these indecomposable ones.
The indecomposable transformations are called ergodic.

0.2. Ergodic theory

Definition 0.13. Let (X,B,m) be a probability space. A measure-preserving transformation T of (X,B,m) is called ergodr,c if the only members B of B with ?-1.B : B satisfym(B): 0 or m(B) :1.

There are several other ways of stating the ergodicity condition and we present someof them in the next two theorems.

Theorem O.L4. If T: X -+ X i,s a measure-preseru'ing transformati,on of the probabi,li,ty
space (X,B,m), then the followi,ng statements are equi'ualent:(i) T i,s ergodzc;(ii) the only members B of B wi,th mlT-rBLB) : 0 are those wi,th m(B) : 0 or*(B) : 7, where T-r BAB i: (T-r B \ B) U (B \"-18);(iii) /or euerg A e B wi,th m(A) > 0, we haue m(uffi=rf-"A) : l;(iv) for euerg A,B e B wi,th m(A) > O,m(B) ) 0, there erists an n > 0 wi,th m(T-"4)B) >0.
Theorem O.LI. If T: X -+ X rs a measure-preseru'ing transformati,on of the probabi,ti,tyspace (X,B,m), then the followi,ng statements are equ'iualent:(i) T i,s ergodi,c;(ii) wheneuer f i,s measurable and (f oT)(r): f (r),Yr € X, then f is constant a.e.;(iii) wheneuer f r,s measurable and (f o:l)(r): f (r) a.e., then f i,s constant a.e.;(iv) wheneuer f e L'(X,m) and 1f oT)(r): f ("),Yr e X, then f is constant a.e.;(v) wheneuer f e L2(X,m) and ff oT')@): f (r) a.e., then f i.s constant a.e.
These are Theorem 1.5 and Theorem 1.6 from Walters [46].
Now we consider a rotation T(r): ar of. a general compact group G. The measure
involved is.the normalized Haar measure mH on B(G).

Theorem O.L6. Let G be a compact group and letT(r): ar be a rotation of G. Theni,s dense i,n G. In part'icular, if T i,s ergodi,c, thenT i,s ergodi,c, i.f and only i,f {o"}P-G i,s abeli,an.

This is Theorem 1.9 from Walters [46].
The following is the well-known Birkhoff ergodic theorem. The proof can be found in
Walters [46, Theorem 1.14].

Theorem O.L7. Suppose thatT: (X,B,m) -+ (X,B,m) i,s measure-preserai,ng (where weallow (X,B,m) to be o-fini,te) and f e Lr(X,m). Then1n'1 \- rrrkt\",tt
nH,, 'tconuerges almost euerywhere to a functzon f* e L'(X,m). Also f* oT - f o."., and i,fm(X) 1crc, then I f.a*: I fdm. Inparti,cular, i,f T i,s ergodi,c on aprobabi,Ii,ty space(X.B.m). then for all f e L'(X.-).

1-I-

|

: I fdm. m-o.e. r e X.
lim :n--+& n ) /(f'(r)7-,J

Chapter 0. Preliminaries

O.2.2 Continuous time
Let (X, B,m) be a measure space.

Definition 0.18. An automorphism of the measure space (X,B,rn) is a one-to-one mapping ? of the space M onto itself such that for all A e B we have TA,T-1A e B andmlA) : mlTA) : m(T-t A).
The measure m is said to be an 'inuariant measure for the automorphism 7.

Definition 0.19. An endomorphi,sm of the space X is a surjective (not necessarily oneto-one) mapping ? of the space M onto itself such that for all A e B we have T-1A e B
andm(A) : m1f-t A1'
Definition 0.20. Suppose {?t} is a one-parameter group of automorphisms of the measure space (X,B,m),t e lR, that is, T'+"(r) :ft(T"(r)) for all l,s € IR and r e X. Then{"t} is said to be a fi,ow if for any measurable function /(r) on X, the function f (Ttr) is
measurable on the Cartesian product X x IR.

The measurability condition appearing in this definition may also be stated in thefollowing (equivalent) form: the mappin1 $: X x R -+ X given by the formula ,lt@,t) :

I'r

dI

1S measllraDle.

Definition 0.21. Suppose {?t} is a one-parameter semigroup of endomorphisms of themeasure space (X, B,m),, € R+ ,: {" : s ) 0}, that is, Tt+"(r) : Tt(T"(r)) for all
f, s € lRa and r € X. Then {"t} is said to be a semi.fl,ow if for any measurable function/(r) on X, the function f lT'r) is measurable on the Cartesian product X x R.a.

We have introduced four fundamental objects studied in ergodic theory: automorphisms, endomorphisms, flows and semiflows in measure spaces. Frrrther the expression"dynamical system" stands for any of these objects. The measure space itself is said tobe the phase space of the dynamical system.

Definition 0.22. The measurable function g is called 'inuariant with respect to the automorphism ? (endomorphism ?, flow {Tr}, semiflow {?t}) if for all r € X, we haveg(Tr) : g(x) : glT-t r)(g(Tr) : s(r),g(T"r) : g(r) for all t € IR,g(T'r) : s@) for all t € R.+).
In other words, an invariant function assumes constant values on every trajectory ofthe dynamical system, that is, an invariant function is a function on trajectories.

Definition 0.23. The set A e 6 is said tobe inuari,ant with respect to the automorphism
7 (endomorphism ?, flow {7t}, semiflow {"'}) if its indicator 1a is an invariant function.
In the case of an automorphism T, the invariance of the set A means that TA: A:

T-rA; in the case of an endomorphism ? or a flow (semiflow) {?t}, the invariance meansthat ?-1(,4) : A or T-tA: A for all t e R (l e IRa).

0.2. Ergodic theory

Definition 0.24. The measurable function g is said to be i:nuarr,ant mod 0 with respect
to the automorphism ? (endomorphism 7, flow or semiflow {"t}), if
S(f ") : g(r) : g(T-rr) almost everywhere@Q") : g(r) almost everywhere,g(T'r): g(r) for any t € R(R+) for almost all r).

Lemma O.25. If g 'is an'inuari,antmod0 functton, then there erists an i,nuariant functi,on!1 such that g - gy almost euerywhere.
This is Lemma 1 (p. 13) from Cornfeld et al. l7l.

Definition 0.26. The set A e 6 is called 'inuariant mod 0 with respect to the automorphism ? (endomorphism 7, flow {7'}, semiflow {"t}) if its indicator l-a is an invariant
function mod 0 with respect to the automorphism ? (endomorphism ?, flow {7t}, semi
flow {7t}).

Definition O.27. A dynamical system on a probability space (X,B,rn) is said to be
ergodi,c if the measure rn(A) of any invariant set,4 equals 0 or 1.

Lemrna O.28. If a d,ynam'ical system'is ergod,'ic, then any i.nuariant functi,on'is constanton any set of full rneasure.
This is Lemma 2 (p. 14) from Cornfeld et al. [7].
Now assume that {gt : t € R.} is a continuous one-parameter subgroup of the abelian
compact group G. Such a subgroup defines a flow {Tt} on G by the formula

Trr:gtt,r€G.

It is obvious that this flow preserves the normalized Haar measure mg.
Theorem O.29. The followi,ng condi,ti,ons are equ'iualent:(i) the fl,ow {Tt} i,s ergodi,c;(ii) the one-parameter group of homeomorphisms {Tt} is m'in'imal, that i,s, for all r e G,
the trajectory {Ttr: f e R.} i,s dense i,n G.
This is Theorem 1/ (p. 99) from Cornfeld et al. l7l.
The following is the Birkhoff-Khinchin ergodic theorem.

Theorem O.3O. Suppose that (X,B,m) i,s a probabi,litA space and f e L|(X,m). Thenfor almost euery (i,n the sense of the measure m) r € X, the followi,ng li,mi,ts erist and areequal to each otherr
Iim lf

n-l

n-l

n

ft:O

k:0

k:-n

t
I
1f
:
tim
Iim
. f /(z-^r.) ::,f*\rJ.
./tr-kr):/(r^r);-l' n-oonL"n-+cxt2n+I/-t"'n--+d)n/-r"i,n the case of an automorphr,sm T; for almost euer!) r € X , the followi,ng li.mi,t eri,sts-t n-lrrrk r)-'J
lim 1\"\ -. f*/\r)'
"\
/-J\r

n'ii'&nk-0

Chapter 0. Preliminaries
i,n the case of an endomorphi,sm T; for almost euery r € X , the following li,mi,ts erist andare equal to each otherri- ] [',]im 1 [' rrr_'r)dr: t-+cn2t1-,,rip I Jo"'[' f e'*,or:'t*-fr+ootJo"y1r'r1or:: f*(r).

i,n the case of a fl,ow {Tt}; for almost eaery r € X , the followi.ng li,mtt eri,sts

ri- I [' y1r'r7or:, f*(r).

t-+cn t Js "i,n the case of a semiflow {Tt}.

Further f.(7"):f.(r) or f*(Ttr): f*(r) wheneuer the ri,ght-hand si,des of theseequat'ions erist. Moreuer

f*e Lr(x,m)and, .tJ
If.o*: Ifo*.
In parti,cular, i,f the dynami,cal system 'is ergod'ic, then f* : I f dm.

0.3 Martingales
This section is taken from Doob [8] and Durrett [9].

0.3.1 Conditional expectation
Let (0, -F, P) be a probability space and let I be a sub-o-field of F. In this section, weonly consider lR-valued random variables. Let Y : (Q, F, P) -+ IR be an integrable randomvariable. Then the conditi,onal erpectat'ion of Y gi,uen Q, denoted byE(Yl9), is defined to
be any random variable Z lhat has(i) Z € 8, that is, Z is Q-measurable, and
(ii)

ff
:
for all A e 8.
Joz(u)dP(cu) JoY(w)dP(w).

It is clear that if the conditional expectation of Y given Q exists, then it is unique P-almost
surely. Let us prove the existence of the conditional expectation. Since Y is integrable,
the function rp defined byrplA) :: I v1r1aY1r1, A € g,Ja

is o-additive on Q and is absolutely continuous to the probability measure P. Thus, accord-

ing to the Radon-Nikodym theorem, there exists a 9-measurable function Z satisfying (ii).
This proves the existence.
Now let {Xt}t.r be any family of random variables. Let g : o(Xt : t € T) be thesmallest o-field of O such that X1's are measurable. Then the conditi,onal erpectati,on of
Y gi.uen {Xr}t.r, denoted by E(f lX1 ,t e T), is defined to beE(Ylg).
Erample 0.37. (i) If y € 9, then E(Ylg): Y.

0.3. Martingales
β(R)and五 ∈g,(li)SuppOSe that y is independent Of g,that is,for all B∈

P({yCB}∩ ■)=P(yc3)P(И

)

Then

E(yg)=Ely].
Here are some properties of the cOnditional expectation.
(i)COnditional expectation is linear:E(α y+zlσ )=α E(ylg)十 E(Zlσ )(ii)If yく z,thenE(ylg)く E(Zσ ).
(面 )If耽

≫O and耽 ↑y wtth Ely]<∞ ,thenE(y72g)↑ E(ylg).

(市 )If 9 iS COn℃ x

and Ely],E[19(y)￨]<∞ ,then9(E(ylσ ))く E(9(y)lg).

(V)If gl⊂ g2,thenE(E(ylg2)lgl)=E(y gl).
(宙 )Ify∈ g and EIZI],El yzl]<∞ ,thenE(yzlg)=yE(Zlg).
(宙 i)SuppOSe

E[ly12]<∞ .Then E(yg)iS the random ttriable Z∈ g that minimizes
the mean square erroノ EIly̲z12].

0.3.2

卜任artingales,deflnitiOnand cOnvergence theOrems

Let{J]2)be a fJι ηαιη,that is,an increasing sequence of sub―σ̲flelds Of F,̀θ■ ⊂ 乃 ⊂ … ⊂ F.
・

We flrst cOnsider the case of real― nlued marting」 e.Let{χ η
N be real― valued random}η ∈唸riables.A sequence{χDeflnitiOn O。

η }おSaid tO be

32.A sequence{」

χη

ααの

サθご

tO{JL}f}れ ∈N iS Said to be a mαχ η ∈ 為fOr al1 2.

ttjη θ α;C(StlpPCrmα

rtじη gα ;θ

stι ttα ttjη gα Jc)(With respect to万 れ)if(1)E[￨」 rη lI<∞,(11){χ π}iS adapted to(′ ]ι },(iil)E(Xη +11J五 )=χ ηfOr all η (Ecrη +11Fl)く Xη Or EcXl+11Jl)≫ χ ηfOr all η).

orChaoter0. Preliminaries

10E"α ttρ Jθ O.33. (i)COnsider a sequence{ξ
with EIξ η]=0・ Let

and為 =ση}Ofintegrable independellt random variablesXη =ξ l+… ・+ξ n,・,ξ れ Then{X72}iS a martingale with respect to{為 }.Indeed,
the Lct that EIlχ ηl<∞ and」Xれ ∈jL are clear.Let us check the condition(111)・(ξ l,一)・

Since attl is independent of九 1,we haveE(X,2+1.為 )=E(χ れら )十 E(ξ η+1ヽ亀 )=χ n+Ela+ll=ス Ъ.

Here we have just used the linearity of conditional expectation and Example O.31(ii).

}Ofintegrable independent random variables with E[ξ η]=1・(ii)COnsider a sequence{ξ れLet」κη=ξ l― ・ξn,and馬 2=σ (ξ l,… ・,ξ π).Then{χれ}iS a martingale with respect to{Fち }・ As above,
we need only to check the condition(ili)・ USing the property(vi)Of COnditional
expectation and Example O.311ii),祀 hⅣ e

E(χ れ+lν L)=E(χ nξ れ+l Jπ )=Xη E(ξ η+lν L)=ス LEIξη+11=ス Ъ・Theorem O。 34.〃 {χれ}づ S αttα 脅焼 θαιθ υ.r.ι .JI αあα ヮ づS a cθ ηυcχ ル ηCι づθη ttlttん
E[9(X?2)￨]<∞ ル r αιιη,ι んθη 9(χ れ)づ S α Sabttα ηθαιθ υ.r.オ .JL.θ θηSθ 9包 θηιιν′ ゲ
p≫ l αηα E[IXη lpl<∞ ル r αιιη,ι んcη lχ η P tS α Sabπ α焼れ θαιθ υ.r.ι .為
7・ rづ

.

J{X72}づ S a Sabttα 電 れ θαιc υ .r.ι .Jh αηα 9 ts αη jη C αSれ θ cο ηυθχ
E19(χ π)ll<∞ ル r αιιη ,ι んθη 9(χη)'S α SZι ttα 滝 れ θαιθ υ .r.ι .JL.θ θηSC―
9包 Cη ιJν ,μ{Xη }tS α Sabttα 付 づηθαιc,ι んθ2(χ η ― α)+tS α Sttbttα rt771θ αιθ′ μ リ グ {χ η}s α θοηsι αηι αηα,s α stlppθ 'ゲ
9篭αrι づηθαιθ′ ιんθη χ η ∧ α ts α sttpρ Crmα jη θαJθ . Hctt α づ

Theorem O。

35。

ル nCι づθη ttlttん

ttι

(χ

η̲α

―α
,0},χ η∧α=min{χゎα)十 :=maX{χ れ}・These are Theorem 5.2.3 and Theorem 5.2.4 from Durrett p].

Theorem O。 36(Martingale convergence theorem)。 J/χ η づS αSab777α 琵れ gα
sup E卜 寸 ]<∞

Jθ

ι
ん
υづ

,

ん EIlχ l]<∞ ・οαι
づmづ ιX υづιιんθη αs η → ∞ ,Xη Cθ ηυcη θs α・S・ ιThis is Theorem 5.2.8 from Durrett 191.
An ilnportant special case of this theorelll is the folloM/ing.

Theorem O.37.J Xn≫
E[X]<E[Xll.

θ,ι んθη αs η → ∞ ,χn→ χ α.s.α ηα0 0S α S切 9permα rι れ θαιTheorem O.38(Zp ma対 mum inequality)。 ιθ̀{χXれ :=15kη

Thenforl(p(oo,C αηαJctη}ι θ α Sabmα 滝 771θ αιχ就.

珊 く(芦 )pЩ χゎ件

Consequently, i,f {Y"} 'is a mart'ingale andY} :: rl&x1gpg2 lYr"l, then到馴 く(芦 )p到 ‰降0.3. Martingales
Theorem O.3g @e convergence theorem). A {X"} 'is a mart'ingale wi,th supEfiX"lr] <n, where p > l, then Xn -+ X a.s. and i,n Lp.
These are Theorem 5.4.3 and Theorem 5.4.5 from Durrett [9].

We now turn to the case of complex-valued martingales. The definition of complexvalued martingale is the same as that of real-valued martingale. Note that {Xn : Un +riV,r), where Un atdVn are real, is a martingale w.r.t. {t"}, if and only if {t/"} and {V"}are martingales w.r.t. {f"}.

Theorem 0.40 (Doob's martingale convergence theorem). Let {X"} be a (real or compler) marti.ngale. Then {lX"l} i,s a submarti,ngale and we haue the followi,ng.

(i) fflim"*- E[l&l] : K < n, then limrr*- Xn:: X* erists a.s. and E[lX-l] < 1{.
In parti,cular, K I @, xf the Xr's are all real and )- 0 or all real and ( 0.

(ii) The followi,ng condr,tzons are equi,ualent:(a) K I n, and the random uariables Xt, X2,.. . , X* const'itute a marti,ngale;(b) the random uariables Xr, Xz, . . . are uni,formly i,ntegrable;(") K{crcandEllx-lJ:K;(d) l( 1n and lirn,,-oo E[|X* - X"l] = 0.

(iii) 1, for some p > 7,lim,rr*E|X"lp] ,'-crc, then th,e condr,ti,ons of (ii) are sati;sfi,ed,
E|X."lP] ,-cn, andJ$n1|x* - x^lal:0.

Conuersely, i,f the condi,ti.ons of (ii) are sati,sfied and i.f EllX."ln] < x-' for some q > l,then

E|x"lnl ( E|x-lql.

This is a part of Theorem 4.1 from Doob f8l.

Part I

Besicovitch Almost Periodic
Functions13

Chapter 1

Besicovitch Almost Periodic
Functions With Fourier E*ponents
Belonging to a Dirichlet Sequence
1.1 Introduction
A function /: R -+ C is called Besicovitch almost periodic if it is a limit of trigonometricpolynomials under some Besicovitch q-(semi)norm (1 ( g < -),ll/lls ': Iimsup

/'t

;T

tl/q

lm J_rttttllodt)

Let Bq denote the quotient space of the q-Besicovitch almost periodic functions (80-a.p.
for short) with respect to the null space Ns : {f t llf llo:0}. Then Bq is a Banach space.
It is well known that Besicovitch almost periodic functions possess mean values and limit
distributions. Here the mean value of a function / is defined as

1rT
Ml/l : ]*m

J_rf

ttat.

provided that the limit exists, and the limit distribution is the probability distribution on
C to which the sequence of probability measuresu7{r:f(r)eA},

AeB(C),

converges as T -+ oo, where z7 denotes the uniform probability measure on [-7,?].
For every function f e Bq, the mean valuea()) : M[/(r)e-i)'] : .ti- :f

1-+@ zt J _7,1ft)e-i^tdt

exists for all ) e R, and those ) for which a(.\) is non-zero are at most countable, called
the Fourier exponents of /. The formal seriesf (t) - fa())e')1I

is called the Fourier series of /. Dealing with the problem of constructing trigonometricpolynomials approximating Besicovitch almost periodic functions in norm, BesicovitchChapter 1. Besicovitch Almost Periodic Functions

16

and Bohr [3] showed that a Bochner-Fej6r sequence of a function / does converge to thefunction itself. Note that a Bochner-Fej6r sequence contains trigonometric polynomials
whose exponents are the Fourier exponents of /.
Let A be a subgroup of 1R.4, the real line with the discrete topology. We consider the
space Bq(A) of Besicovitch almost periodic functions whose Fourier exponents belong toA,
Bq(A) ,: {./ € Bq : a(\): 0 if / A}.

^
Let I be the dual gro^up of A and z be the normalized Haar meas.,r" or. I. Then thespaces Bs(A) and -Lq(A, z) are isometrically isomorphic under the isomorphism 7n whichmaps a Bq-a.p.function

f(t)-!a())e')',\€A

to an Lq(i,z) functio"Tq(f) with the usual Fourier series

4(/)(r) - fa())11(r).

^€Awhere for A e A, x.r(r) : r()), (z e [) is a character of I. Moreover, the ]imit distributionof / coincides with the distribution of Tq(f)The main aim of this chapter is to study Besicovitch almost periodic functions whose
Fourier series are of the forms

λ
C→ ″
れ
∫
0〜 Σ α

(1・

,

1)

m:I

where {,\-} is a strictly increasing sequence of non-negative numbers tending to infinity,
called a Dirichlet sequence. Let A be a subgroup of IR4, generated by {)-}, and let A be
the dual group of A with the normalized Haar measure v. Then the limit distribution of/ coincides with the distribution of an trq(fr, z) function

πχ λ
m(″
■(ノ )(″ )〜 Σ α

)・

(1.2)

For q ) 1, we will prove in this chapter,a":;" Fourier series (1.1) converges in norm withthe usual order, which gives another way to approximate this kind of Bq-a.p.function bytrigonometric polynomials. Equivalently, by the isometric property, the Fourier series (1.2)
converges in ,Lq(A, z) with the usual order. In addition, we will show that the Fourier
series (1.2) converges almost everywhere (with respect to u). This result is analogous
to Carleson's theorem for classical Fourier series on l},2trl, and in fact is a consequenceof Carleson's theorem in multi-dimensional case. However, to apply Carleson's theorem
to our present case, we need to introduce another way to identify limit distributions offunctions of the forms (1.1). as we will see later.

L.2 General theory of Besicovitch almost periodic functions
)C .λ

ι

]=

￡

(ι

１一
π

α(λ )=NIIIノ

ｍ
¨
ｉ
︲

Recall that if a function f is Ba-a.p., then the mean valuef (t)e-i^t dt

ワｒ

1.3. Probabilitv sDace associated with Besicovitch almost periodic functions

alues ofexists for all real、 zalues of λ,and is non― zero only for at inost a countable set of、 ア入,say,
入1,入 2,… …

Let α
222=〜 lI∫ (ι )C 'λ mtl.Thenf (t)-T o^"'^^r

is the Fourier series of /.
Let us introduce an algorithm for the approximation of / by trigonometric polynomials 13]. If p's are Q-linearly independent numbers, we consider Bochner-Fej6r polynomialsσ(移 ll移 :IIlβ l)(ι・
(1‑響 )… (1‑響

)=

)α

m♂

λ
″

、
vhere

入れ=ν lβ l+・ …+r/pら

,

and α
m is to be interpreted as zero、 vhen the above linear combination of β's doeS notbelong to the Fourier exponents ofノ .We Will use the notation

notationσ3(ι )inStead of the detailed

σ惚
樅 D°・

=β
Let αl,α 2,… ・ be a sequence ofく 2‑linearly
independent positive numbers(whiCh gen―erally is ininite but in particular cases mtt be inite)such that every exponent入 772m叩

α's with rational coemcients,

be expressed as a inite linear form in the

λm=r772,lα l+rm,2α 2+・ … 十 rm,9れ α9れ
石
ヽヽ
e put

=赤
l=器 2=蒜 "… た
β
,β

,β

.

,

where Arl,」 v2,… ・,ゴ vた are pOSitive integers.The result on the approxillnation of B9‑a.p.

fllnction by Bochner― Feier p01ynOmials is as follows.

Theorem l。 1(p,Theorem I珂
9‑η θrm,α sた → ∞Rθ ttα rt・

,M→

)。

ι
cん
んθBcs,cθ υづrん CS鶴 鶴 σ3(ι )6θ ηυθη θS tθ ノ(t)づ η ι∞ ,乃 → ∞ ,… .,α ηα

齢→ ∞ ,赫 → ∞ ,… …

1.2.A sequence of Bochner― Fei6r p01ynOmials
σ31(ι ),σ 32(ι )'… ・

is called Bochner― Fej6r seqllence if the basic numbers,電 たβl,… .,β たand the indices 21,… 。

satisfy the COnditions of the above theorem.

1.3

Probability space associated with Besicovitch allnost
periodic functions

l.3.l

Besicovitch functions whose Fourier exponents belong to a sub―group

Let A be a subgroup of Rd,the real line、e consider the spacevith the discrete topology.ヽ ヽ石

B9(A)of BeSiCO宙 tch almost periodic fllnctions whose Fourier exponents belong to A,

39(A):={ノ ∈B9:α (λ )=O if λ￠A}=the linear closure of{CJλι:λ ∈A}with respect to the ・￨19 norm.

Chapter 1. Besicovitch Almost Periodic Functions

18

The second identity follows from the convergence of a Bochner-Fej6r sequence to thefunction^itself (Theorem 1.1).
Let A be the dual group of A. Since A is discrete, it follows that A is a compact abeliangroup. Thus, thele is a unique norma"lized Haar measure u on A. For ) e A, let X1 denotethe character of A which maps r e A to r()),X^\r): r()), r e f.
Then {11}16n ar€ the characters of I.
For t € lR, let e; be the element of I defined by
e1()) : six' '\ e A'
Then the mapping R. ) I F+ et ei is continuous, and hence is Borel measurable. Consequently, probability measures ur. ott (R,6(R)) induce on (A,6(A)) probability measures

Qr(A):ur(r i er € A), AeB(i).

ProoJ. For any character X.\, (tr e A) of fr, it i, clear thatl:,ys(e)dt:

={学

,￨

ご

Ｃ

:+

λ

l^xx@)aar@)

ぽ
﹄
Ｈ判
割

Theorem L.3. The sequence of probabi,li,ty measures {Qr} conuerges weakly to the normali,zed Haar measure u as T -+ m.

→ス
λ
χ
Oご ν
。as 7-+m.
)

Since the linear space spanned by the characters {11}16n is dense in C(I), the spaceof continuous functions on A, the assertion of this theorem easily follows. (See also thecontinuity theorem for probability measures on compact abelian group (Theorem 0.11).)

If A is dense in lR., then {"r}r.n are distinct. The only subgroups of IR that are notdense are isomorphic to the additive group of the integers. In this case, A is the classicalcircle A : {z e C.: lzl:1}, et: €t+To for some Ts } 0, and {e1}1.p : 1.

Lemma L.4. Unlesr fr ,r a ci,rcle, the characters {e1}16p are dl,stinct and, form a d,ense
one-pararneter subgroup of l\.
Proof. Let ,9 be the closure of {e1}1sp. Obviously. Qr(S) : 1 for any T > 0. Since theset ,9 is closed, it follows from the property of weak convergence thatu(s) > t'ff:o er(s):1.

Thus, the complement of S, S' : fr \ S is open and has z-measure zero, which impliesnthat ,Sc : 0 by the property of the Haar measure. The proof is complete.

Theorem L.5. Tt, [+ [,1i e R), d,efined, bg Tt(r) : r,€tt i,s an ergod,ic fl"ow (wi,threspect to the Haar measure u).

1.3. Probabilitv soace

19

Proof . We need to verify the measurability of the flow {Tt}16p. It is clear that the mapping/:lR.xfr-+fr(t,r) r+ e,rｆ □ｏ

is continuous (with respect to the product topology on n x l). Hence ilis 6(R) I
B(L)lB(L)-measurable. Consequently, for any measurable function f (r) on A, the function f (Ttr) is 6(R) I6(A)-measurable.
The ergodic property of the flow {Tt}rep, which is equivalent to the denseness{"r}r.o in A, follows (see Theorem 0.29).

We construct an isometric isomorphism Tn: Bq(/\) -+ .Ls(A,z) as follows. For a
trigonometric polynomial p(t) whose exponents belong to A of the form'o())e'"'p(t) : z-2
)
,
\._

,

i\+

(I t finite subset of A),

ncl

we define,s-Tq(p)(r) : >a())11(r) € c(A) c Lq(1\,u).
)€1

Then it follows from the property of weak convergence thatlfTlfT|X o J -rlnft)lq dt : ]tyLn J -rlrn{n){e1)lq dt: ,19_ f-fJ^lrn@{r)lqdQrQ)

and hence,llpllq:J^lro@)Q)lqdu(r).

: llrn(dll r, 6,4.

(1.3)

Thus, Tn can be continuously extended to an isometric mapping from Bq(A) 1o tru1i,u).
Indeed, Iet / € gu(l\).Then there exists a sequence of trigonometric polynomials {pn}t,whose exponents belong to A, which converges to /, that i., llpr - f lln -+ 0 as ,k -+ oo.
In particular,{p!"}n is a Cauchy sequence in Bq(A). By (1.3), {Tn@*)}* is also a Cauchy
sequence in trq(l,u), and hence, the limit 1im6;- Tn@i exists in trq(I, u). LetTn$)::1im6;- Tn@,r). It is clear that Tn is well defined; moreover) it follows from (1.3) and fromthe continuity of norms thatll/lln: llTq(f)llrnr.[.,r. f e Bs(A).

(1.4)

Ъ is illiectiVe.on the other hand,■ iS SuttectiК Since the linear spacespanned by the characters{χ λ
A iS dense in Z9(κ ,ν )臭 nd B9(A)is complete.Therefore}λ ∈This implies that

■ iS an iSOmetric isomorphism between 39(A)and L9(A,ν ).
Theorem l.6.Fθ r lく 9<∞ ′BC(A)笙 Z9(A,ν ).
TheOrem l.7(117,Correspondence Theorem])。

T/1θ tsθ mθηtsm■ ん
α
S ι
ん
θル υづ
θ
ηん

pttρ θ fjcs.
7・(1)写2づ Sα Jれ θαr tsθ ttθ trづ θ鶴 鋼9Pじ ηθルリ鶴 B9(A)θ ηιθ Z9(A,ν ).
(ii)■ (∫ )=7し (∫ )・;JθChapter 1. Besicovitch Almost Periodic Functions=1ル

ごノ∈Bp(A),g∈ B9(A),ι んθ
T's tη θιづιηЛυJα θ+1/9(γ ≫ 1),α η
9包 αν(iii)f/1/γ
づ
mpJじ csノ g∈ Br(A)α ttd
写(力 )=写 (ノ )■ (g).
んαS tん θ ttarづ θr scrJcs Σ λ Aα (入 )♂
∈(iV)Jノtん cλt,ιんθη ■ (ノ ),α S α ル ηCtづ θη じ
η L9(A,ν ),ん αS

Foarじ cr scれ cs鰺λ。
■げ)0)〜 Σα
)χ

)・

λ∈A

f7t pα rι jθ ttι

αr,MF]=EI■ (ハ

づ
鶴づ
ιご
づ
strづ
(V)動 θιbttι jθl.

ηげ∫Cο れCtacs υづιんιん
cα づれ じ
ο
ηげ■(∫
sι

bttι

)・

Pη げ (う and(i五 )are Clear.(五 )eadly f0110ws iom the inequ」 ity‖ αl― lbll≦ lα ― bl fOr
any α,b∈ C.(市 )f0110WS from the Lct that the conヽ 配rgence in BC or in L9(A,ν )implies
the convergence of each Fourier coemcient.

To prove(v),let∫ ∈B9(A).Then there is a sequence of trigonometric polynomials{pη }WhOSe

exponents belong to A converging toノ

浮with respect to the ll・ ￨19 nOrm,sa)Ъ

ノ￨19=0.

=LII%―
Let θT:(Ω ,P)→ [― T,T]be a random variable unifOrmly distributed on[― T,T]。 We put
XT,れ :=pn(θ T).

Note that the distribution of χT,72 is the prObability lneasure r/TP万

1. Thus,by the lnapping

theOrem(TheOrem O.3),
xT,πム ■(pπ )aS T→ ∞,(1.5)

because■ (pη )iS COntinuous.Moreover,by theお omet五 c property of■ ,■ (pn)conVergestO場 (ノ )in L9(A,ν ),whiCh implies thato.
場(pη )ム ■(∫ )as 2→ 。

(1.6)

In addition,for any ε>0,using Chebyshev's inequalit,lM″ e have

―
η
玲 く
歎
陶β
レ
O― ノ
0隣
:寡 ￡
￨>→

,

whereン 争 :=ノ (θ T),and hence,

1雪LI巴 p歎 陶 β―珊 ≫ → く :漂 コLレ π一洲 1=Q

田

Therefore,by using Theorem O.7,lve get the desired result

玲 ム ■(∫ )as T→ OO・□

ＯＺ

1.3. Probability spacet.3.2 Besicovitch functions whose Fourier exponents belong to a Dirichlet sequence
Consider a Dirichlet sequence {)-},and letBo({A^}):: the linear closure of {e-i)-t1 with respect to the ll.lln norm.
Let A be a subgroup of 1R4, generated by {)-}, and let I b" th" dual group of A with thenormalized Haar measure u as in the previous subsection. Then the limit distribution ofa function f e Ba({\^}) with the Fourier series

.f(t)

- m:l
f o^r-'^^'.

coincides with the distribution of an trq(fr, z) function- i. o*r-^* e Lq(i,u).

TnU)-q\J /^-t,

We now introduce another way to identify distributions of functions in Bq({)-}). Let? : {s e C : lsl : 1} be the unit circle on the complex plane, and let

o: fr r-,tn:lwhere 1rn: j for all m € N. With the product topology and pointwise multiplication, theinfinite-dimensional torus O is a compact topological abelian group. For T ) 0, we define
a probability measure Qr on (0,6(CI)) by

Qr(A) :: ur(r : (s-ir\^ )-ex € 1), A e B(O).
Let

z: @z*.
m€N

where Zrn : Z for all rn € N. Then the dual group of O is isomorphic to Z and the dualgroup of Z is isomorphic to Q. Each k: {k^: rn e N} € Z,where only a finite numberof krn are non-zero, is identified with a charactera r+ wh : flw(m)k*of O. Here c-.,(m) is the projection of u € f) oiro ,-.Conversely, each 0r € O is identified
with a character 1k r-+ uL| of Z.
Recall that for a probability measure Q on (Q,S(CI)), the Fourier transform 9(a)(k) ofis
Q defined bys@)&) ;: B(Q) tb):; (+ a(m)k^) oo, k e zThen we construct the probability measure P on (O,6(fl)) as follows.

(1.8)

Chapter 1. Besicovitch Almost Periodic Functions

22

Theorem L.8. There i,s a probabi,h,ty measureP on (O,B(f))) such that the sequence of
probabi,li,ty measures {Q7} conuerges weakly to P as ? -+ oo. The Fouri,er transf orm of P
rs gzuen by

Dfr:r\,nk,n:6,^,,-, - l',vtrr-\0.
dL;^
tft^^k-+0.

Moreouer i,f {^r"} i,s Q-lznearly i,ndependent, then P coi,nci,des wi,th the normal'ized Haar
measure on dl.
Proof . The Fourier transform gr(b) of the measure Q7 is of the form\t tr/*
gr(k): / (|| wk^(m)loer:+/_(f[,-itx^*^ls,zt J-rJo \,"^i
/
/
\i-1
( t,
if ;;ff:, \qk*: Q,: { exp{-irDfr:r\*k*} - exp{eTlff:,+ 6.
^mkm\ . if )-r_:.\,,.k,.
\,,k^
2ir
|.
Dn_1
r/*

\

I

Hence, the limits(k)::,h g,(r) :{;: ll 3$:l }:i:rz:,

exists for every k. The continuity theorem (Theorem 0.11) implies that there exists a
probability measure P on f) such that {Q7} converges weakly to P as 7 -+ oo. Moreover, 9(k) is the Fourier transform of P. Now, if {)-} is Q-linearly independent, thenDff:, \^k-: 0 holds iff k,n: 0 for all m. It then follows that P coincides with theInormalized Haar measure on f,).

Lemma L.9. {w(m)}-ex zs an orthonormal system in L2(Q,P), that i's,E(P)[,(m ,),(-,)]: {1. {,:,")T,i:

Here E(P) denotes the erpectati.on wi,th respect to P.
Proof . Let m1 I *2. Take ! : {k^: rn € N} such that k-, :are zero. We have Dff:r \*k^: )-, - \rn, f 0. Therefore

7,k*r: -1 and the others

E(P)lr1tn1 )qntr)l: g(k) : 0.
which completes the proof.

The following theorem is similar to Theorem 1.7(v).

Theorem I.LO. Let f e Bt({),,"}) wi,th the Fouri,er seri,es/_t

aTns-i^-t.

Then the sequence oJ probabi,ti,ry *"orur"T

u7(r:f(r)eA), AeB(C),conuerges weaklg to the distri,buti,on of f(r) - D^a^u(m) € ,s(O'P). Here f(w) l*a,nu(m) means that f belongs to the l'inear closure of {u(;m)}^6s 'uzth respect to the
Lq(Q'P) norm ando^: I f (u)u(m)dp(c.,,).
JO

L.4. Converqence results
We are now in a position to characterize the dual group of A and the support of P.
Recall that A is a subgroup of Ra generated by {)-}. Hence, the mapping
to: Z -+ ll,

A.: (A.,n) ++ - L^^O^
is an onto group homomorphism. Thus by tnl nr.t isomorphism theorem in abstract
algebra,

L= Zlkercp.
Here ker g:{k e Z:l^\,nk,n:0}

is a (closed) subgroup of Z. Let

K : {w e O: cu&: 1 for all k e kerrp}.
Then 1{ is a closed subgroup of f), called the anni,hi,lator of ker g. The result on the
duality between subgroups and quotient groups ([42, Theorem 2.7.2)) gives the following.

Theorem l.lL. The dual group of lt i,s isomorphi,c to the subgroup K of Q.
Since K is a closed, and hence a measurable subset of 0, the normalized Haar measureu on K can be regarded as the probability measure P'on Q defined byP'(A):: u(An q.

A e B(Q.

Let us now calculate the Fourier transform of P'. First, by the duality, ker g is theannihilator of 1{ because l( is the annihilator of kerg (142, Lemma 2.1.3]), or ker g canbe rewritten as

kerp : {k : wh: I for all cu € K}.

This implies that a character ft of O, restricted on K, is the trivial character of 1f if andonly if k e ker cp. Thus we obtain the Fourier transform of P/,e(p,)(&): IrLae,: [#ar:{|.

Jx

Ja

if ke kere.

|.0. otherwise.

It follows that P : P/ because P and P' have the same Fourier transform. Moreover, by
the property of Haar measure, supp(z) :above, we get the following.

K. Hence supp(P) : 6. Combining all the

Theorem 1-.1-2. (i) supp(P) : I{.
(ii) Plft' : u, the normal'ized Haar measure on K.

L.4 Convergence results
I.4.L Classical Fourier series
Let trq([O, 2tr)) be the space of q-th power Lebesgue integrable functions f : l0,2tr] -+ C
endowed with the normr,
/ 1 r2r:(ll/11r,,110,2"1;; I ltt"l!a"1\zr Jol/q/

For / e Ls(P,2rl), the Fourier coefficients an are defined by the formula
2π

α
η==,ア

ズ

η
″
∫(″ )C を d", η∈Z.

Chaoter 1. Besicovitch Almost Periodic Functions

24

Then the seriesmf(r)- tis called the Fourier series of /. For nO a^,

Srr(/)(") :anein'*lt

anein'.

lnl(NTheorem 1.13 (126, Theorem II.1.5]). For 7 < Q ( m, the parti.al sums 571(f) of any
f e Ls(l},2trl) conuerge to f i.n Lq(l},2rl) as ly' -+ oo.
This theorem is equivalent to the following.

Theorem l,L4 (126, Theorem II.1.1l). For 7 < Q ( oo, there r,s a constant Kn such that

foranyfeLq(l},2rl)
||

S,^,'

(/) ;c 11o,zrl) < Kn ll / ll r' ([o,zn]) .
||Indeed, let S,r,, be a continuous linear operator on.Lq(10,2n]) defined as / -+ S,^r("f),
whose operator norm is denoted bv llSrollq. Now if the sequence {Sry(/)} converges to /,then {S1,.(/)} is bounded for every / € ,s([0,2n]). Therefore, {llS,r,'lln} is bounded by the
uniform boundedness principle. Conversely, assume that there is a constant Kn such that
forany f eLq(10,2n)),||S,^r ( / ) ll rn qo,z^11< Kn / z,' [0,zzr] )
I |

||

(

.

e Lq([0,2T]). Given any € ) 0, there is a trigonometric polynomial P(r) :Dnb&ok" satisfying llf - Pllr,<to,znl) ( el(Kn + I). For N greater than the degree ofP, we have ,Sry(P) : P, and thus
Let fllSr,,(/)- fll7op,ur1; : llSru(/) - Sru(P) + P - f llrn{o,z"y
E lls,^r(/) - sru(P)ll7,e 11o,zrl) + llP * flluqop"1l
<'

: s.
Kr=:
"Kn+I' + =-1
Kq+1

which completes the proof of the equivalence between the above two theorems.
Carleson [6] showed that the Fourier series of an tr2([0,2n]) function converges almost
everywhere. Later on Hunt [24] generalized this to .Lr([0,2n]),Q < p < oo). This result
is now known as Carleson's theorem or the Carleson-Hunt theorem.

Theorem 1.15 (The Carleson-Hunt theorem). For 1 < q < m, the partial sums S7'r(f)of any / e la(fO,2nl) conuerge almost euerywhere to f as N -+ oo.

'

To prove this, we consider the maximal function

Nn/(")': i,l;; lsrr(/)(r)l.

The almost everywhere convergence is a consequence of the following maximal inequality.

Theorem 1.16 ([6, 24]). For I < q ( oo, there zs a constant Cn such that for ang
f e Lo(l},2nl), we hauellNa/ ll'tlo,r"r) < Cs ll / ll ro (o,zr))'I.4. Convergence results
The multi-dimensional version of this maximal inequality was investigated by Fefferman [16] . For our purpose, we only mention a special case of Fefferman's result. Let
f e Lo(l},Ar)d) with Fourier coefficients {an}bezo, where d e N, d> 2 being fixecl. For
u : (ut,. . .,ud) € (R+)d, we consider the maximal function

N{ro/(r)': :lB I _R, orei{&'r)1.

(&,q)<b

Here N* : NU {0} and (r,g) denotes the inner product of r and y in JRd,d(r,a):L"trt,J:

L: (rt,...,ra),a: (at,...,gd) € Rd.

s'

Lemma 1.17 ([16, 22]). For 7 < q { oo, for any 9:f € Ls(l},2nlo),(ut,...,ud) € (R+)d and, anyllMIs,/ llz,, t ll,2rld) ( Cq ll / llr t lo,2r)d),where Cn i,s the constant i,n Theorem 7.76.

L.4.2 Convergence of Fourier series of Besicovitch almost periodic functions
Recall that supp(P) : X is isomorphic to the dual group of A. 1{ itself is a compact
abelian group and its normalized Haar measure z coincides with Plsupp(p). A summablefunction on K has Fourier series

f (w) - |atr)t.r(cr).

)eA

For q ) 7, let Hq(K, {)-}) (or Hq(fr, {l-})) be the subspace of La(K,u) : Lq(9,P)
consisting of those functions / whose Fourier coefficients ci,()) are zero except for .\ e{-)r,-12:...}. A function f e Hl(K,{^-}) has Fourier series of the formf (r)- m:lio^r-^^: rn:li a^u(m).

Next, we will establish the maximal inequality for functions in flq(f), {,\-}). LettM

N4/(r) :: sup lf
MroI

o-r(rn)l

lEr

.

I

Theorem L.I8. For I < q < a, and for any f e Ha(Q, {)-}),
llNn/ ll z,' ro,p t < C qll f ll re(o,P),

where Cn i,s the constant i,n Theorem I.76.

We need some preliminary results before proving Theorem 1.18.

Lemma L.19. Let M e N and let {)-}r<-<,lz be positi,ue numbers. Then there are
Q-li,nearly zndependent posr,ti.ue numbers {po}r<o<" such that{),n}r(m<M C O

1(p(P

N*pp.

Chaoter 1. Besicovitch Almost Periodic Functions

26

Proof. We prove by induction on M. Of course, there is nothing to do if M : l. Assume
that this lemma holds for some M ) 1. We now prove it for M * 1. Let {)-}1g-g,vr1r bepositive numbers. By the induction hypothesis, there are Q-linearly independent positive
numbers {pr}r<o<l' such that{tr-}r<-<,y c O

1(p(P ^.ro.

Case 1: )na11 is Q-linearly independent with {po}r<o<". Simply set pp11 : )14111, w€
obtain the desired result{)-}r<-<,nnr+r C

N*pp.

O

1(p(P*1
Case 2: )ya1 is Q-linearly dependent with {pr}1gp(pi namely there are rational numbers{ko}r<o<p such thatp\nr+r:Dror,
P:t

Let Pt : {p t ko > 0} and Pz : {p : k, < 0}. If P2 : 0, then\w+t e O Q.p,o,1(p(Pwhere Q* : Q+ U {0}. Thus with a suitable number I'I € N, by letting p;:(IlN)pp,p:1,...,P,weget)1a11 €O X.r;.

1(p(PClearly,{)-}r<-<,y c O

1(p(P ^.r;.

Therefore this lemma holds for M +7 if P2:0. Next we consider the case PzlA. Let
p2 € Pz. It is enough to construct positive numbers {Ai}rge<p that( { r,',1r"r", is Q-linearly independent.

{ {)-}r<-<,,r C Or<p< pQ*tt!p,l.)nrnr :t'r:rt ;lo, ki . Q, (p : r,...,P); PL : {p t kL < 0} :Pz\ {pz}.
To construct {pi}r<o<p,let {rr}r6p, be positive rational numbers satisfying(r-!/:p€Pr'P:-kTr'

t0.ro lkp4plFpr. pePt.

The existence of {rp}pep, is ensured because

0()u+r: t kppp*Lrrro< I
p€Pr
P€Pz

Let

,

kp4plkp,tf,p,.

P€Ptlro-ftofkp)Hoz. PePr.
l.Po' P/Pt'

Fp:
'P \

(1'9)

″
９ん

1,.4. Conversence results
Clearly, {pi},<o<p is Q-linearly independent and{Pp}r<p<p

c O O-r;
1(p(P

Thus

{)-}rs-g,vc O *-ro. O O-r;.
l(p(P

l(p(P

Let us consider the representation of )aa+r with respect to {p,|}srap,),v1+r: D krro+ I

kp4p

p€Pt

p€Pz

p€Pt

pePz\{pz}:tkppL+ t: D *rfuo - Qolkr)pr,) + t
pePt

kp4p

pePz\{pz}krioHence the sequenc" {pl}r<p<p satisfies the condition (1.9). The proof is complete. I

For each ,&1 e N, let Q7,1 : [IX:r7- (with the product topology and pointwise
multiplication). Ona is also a compact topological abelian group. The projection prpl
from 0 onto f)y is continuous. Let P 1e,1 :: P o pr-n| . We can check that
9u(ktu)1(M\ (kr': !t' it DX:'\Ynk^: Q' k(u)'l- '|, ' ' ' 'knt) ',$"ru^'[0, it Dx:r^^r* +;:,

is the Fourier transform of P1a.
For the sequence {tr-}r<-<,1.r, Iet {pp}rgp<p be positive numbers as in Lemma 1.19
and let {k^o} c N* be the coordinates of {)-}r<-<,lr with respect to {pp}r<pgr, that is,P)- : I k^pFp, (m: L,..., M).
P:I

Let f2! : lIo:r7o (with the product topology, pointwise multiplication) and let m| bethe normalized Haar measure on O'". Define jvt Q'p -+ 01a asP(r'(p))r<o< p ,+ (u(m))r<-<,r2, u(m) : n r'(p)o*, .
P:7Then the mapping j y is also continuous. The following lemma easily follows from thecalculation of the Fourier transform and from the continuity of jy.

Lemma1.20. (i) PM:^'p" j-t.
(ii) supp(P u) : jy(Q'p).
Proof of Theorem 7.78. Step -1. Assume that only a finite number of {a-} are non-zero)that is, an M € N exists such that em:0 if m > M. Since the maximal function in thiscase only depends on the first IV coordinates, in what follows we identify Q,14 with Q. For
each k : (kr, ...,kp) € N*P, Iet
.. sP

tt
kpFp: )- for some rn) (1 ( - < M)'
or: l1^'
Ptl
0, otherwise.
l'|.

Chapter 1. Besicovitch Almost Periodic Functions

28

Then when ω=」 llf(ω ′),We have that for any L∈ N,

ω
mω い
)=Σ α
と
・
Σα
,

m<L

ん∈N*P

(と ,μ )<λL

where μ =(μ l,… .,μ P)・ Therefore

M力 MOつ )=11:￨ん

A・

04・

IPα

(1,μ )<bdimensiOnal maximal inequality(Lemma l.17)that

lt follows frOm the multi―′
Mlノ χ(ω ))￨lz,(Ω
‖
)く ら
Ъ卜
(プ

′

￨￨∫ (プ,れ卜M(ω ))L9(Ω ,mり.

NIIoreover,by Lemma l.20,v′ e have

■ (ω )￨lzq(Ω M,PM)=‖ M∫ (力 r(ω り‖llノ)￨￨五 c(Ω′,mりЪノ(ω )L9(Ω flr,PM)=￨￨ノ (プ M(ω ))IL。 (Ω 3,mЪ
‖

,

)・

TherefOre
C91ノ (ω )I L9(Ω M,PM)・‖ヽ 1ノ (ω )IL9(Ω M,PM)くStcP 2. 1ヽ 石
e no、 v prove the maxilnal inequality in general case. For each if∈

ノ f‰ αぃM岡 0=。

1屁mω

Then MII(M)ノ (ω )inCreasingly converges to M∫ (ω )as y→‖Mノ ￨IL9(Ω ,P)=爵

吼

N,let

)「

∞ ・ COnsequentlェ

‖M(M)∫ L9(Ω ,P).

Applying Step l to a fllnction g(ω )of the form

gO)=Σ E αωれい)十 1E bπ ω
い

),

m=l

m=lf+1■ and{bれ }being arbitrary,we obtainf′'Mθ (ω )L90,P)く の lθ (ω )￨IL9但 ,P)・‖Moreover,for g(ω )Of that fOrm,祀

hNe M(M)ノ(ω )く‖M(M)ノ (ω )￨IL9(Ω ,P)く ら

Mク (ω ),WhiCh implies thatg(ω )￨IL9(Ω ,P).

By choosing a sequence of g(ω )COnverging to∫ (ω )in z9(Ω ,P),We arrive atM(M)∫ (ω )L9(Ω ,P)く ら‖
The proof is cOnaplete by letting̲llf一 → oo.

￨∫ (ω)L9(Ω ,P).
□

29

1..4. Convergence results

Theorem L.21. For 1 < q < @, Lt)e haue the followi,ng.
(i) For any f (t) -la*e-ix^t e Bq({ )-}), the parti,al sums
M

Su(f)(t): t aps i^^tconueT-ge to f in the ll.lln norm as M -+ m.
(ii) Equi,ualently, for any f (w) -la*w(m) e Hs(O, {)-}), the partial sums

M

Su(f)(r): I

a,nw(m)conuerge to f i,n Lq(Q,P) os M -+ a.

Theorem 1.21 is a consequence of the maximal inequality (Theorem 1.18) and thefollowing result whose proof is similar to the proof of the equivalence between Theorem 1.13
and Theorem 1.14.

Lemma L.22. The followi,ng two condi,ti,ons are equiualent:(i) for any f (w) -la^w(m) e I{c(f,), {)-}),
M

Stt(f)(r): t

a-w(m) f\t as M -+ n;tn:l(ii) there'is a constant Kq) 0 such that for anA f e Hq(Q, {)-}),
lls u (f)ll u ro,P) < K qll f ll r., p,pt.

Theorem 1.23. ForI < q ( oo, andfor any f(u) -\a,,u(m)parti,al surris Sp1(f)(w) conuerge P-a.e. to / as M -+ o.

e fla(fl,{),*}), theProof. By applying Theorem 1.18 to the function f - SuU), we obtainll-tl< cnlll - su(f)llz,n1c,,ey.
ll sup ls-(/) - sr17;1;1'" 'll^>trllz,ntsz,pt

Thus sup->mlS^(f) - Su(f)l converges to 0 in LI(Q,P) as M -+ oo. It follows that
there is a subsequence {Mr} c N such thatsup lS,,(/)(r) - 574rff)@)l -+ 0 as ,k -+ oo for P-a.e. ar € fl.

m)-Mn

Hence {^9,u(/)(r.,)},rz is a Cauchy sequence for P-a.e. c,., € O. Consequently, the limit

f'(r): nlrm Su(f)(a)exists for P-a.e. ar € CI. On the other hand, the function / is the trq-limit of {S,y(/)}(Theorem 1.21). Therefore / : ft for P-a.e. c,.r € f), which completes the proof of Theorem 1.23.
IChapter 1. Besicovitch Almost Periodic Functions
1。 4。

3

The linearly independent casem}iS Q― linearly independent and give anotherprooお of Theorem l.18,Theorem l.21 and Theorem l.23.Assume that{入 m}is Q― linearly
independent. Then the probability lneasure lP coincides M″ ith the normalized Haar ineasureln this section,we consider the case when{入On Ω.Under P,the sequence{ω (971)}becomes independent.

P印 げげ 7Lθ θ
ttm l.21 αηαT77θ θ nl.23.Let
9(Ωmω (m)∈ ∬,{入 π∫(ω )〜 Σ α
}).
π =1

Then{Sllf}M∈ N iS a martingale with respect to the■ ltration
九 y=σ (ω (1),… .,ω (■ f)),because{ω (m)}iS a Sequence of independent random vttiables with means zero,where
SM:=SM(ノ ).On the other hand,■ 対 ng if,we consider{】 乞 }L>lf Of the form
rlf

均=ΣE αmω ぃい),)十Σ L bれ ω
π =l(1・ 10if<mくwhere{bm}iS an arbitrary sequence of complex numbers.Then{ン ■}L>Ar iS also a mar―tingale,and hence,{yL19}L>M iS a submartingale.Consequently,
E(P)[￨ン ■91.

E(P)[ISM 91=E(P)IlyAf 9]くSince there is a sequence of{ン 乞}Of the above form which conヽ 配rges to∫ in Z9(Ω ,P),it
follo、 vs that

E(P)[SM 9〕 くE(P)[ノ 91<∞ 。

(1・ 11)Thus,if 9>1,then by Doob's martingale convergence theorem(see Theorem O.40(面 )),9(Ωthe sequence{島 7}COnverges P― a.e.and converges in ι,P)tO a random ttriable S∞.

It is clear that
S。。(ω )〜

Σα222ω (m)名配 ==1is the Fourier series of S∞ .This implies that S。 。=ノ ,P― a.e。 ,which complete the prooおof Theoreln l.21 and Theoreln l.23.
Pη げげ T72Cθ ttm l.18.Let□

端 :=1群 動 1銑￨・Then by L9 maximum inequality(see Theorem O.38),we obtain

EIPl聞 判 く

動(芦 )9ゴド 冽 判く助(芦 )9ゴⅣ 円 <鳴where the secOnd inequality follows from(1.11)・ Note that S揚 (ω )increasingly convergestO ⅣI∫ (ω ).COnsequcntly

ゴDⅣ 川判=爵吼

ゴD聞 判く(芦 )9ゴ動Ⅳ昨□I.4. Convergence results

31

Theorem 1.24. Let f e H' (0, {)-}) wi,th the Fourier series f (r) - Iil-r a^a(m).
Then the parti,al sums {Sy,a(/)(r)} conuerge P-a.e. and conuerge rn Lt (CI,P) to f as

M-+n.

Proof. Similarly as in the proof of Theorem 1.21 and Theorem 1.23, we haveBre)1swlJ ( B(prll/ll < _.
Moreover, taking the sequence {Yr} of the form (1.10) converging to / in rt(Q,P), thenthe sequence {E(P)(rrlfd} also converges in tt(CI,P) to p(P)(/lfir). Consequently,Ee)gfM): snr'
This implies that the sequence {S.'rz} is uniformly integrable. Therefore, by Doob's martingale convergence theorem (see Theorem 0.40(ii)), {,Sy} converges to a limit ,9"o almost
everywhere and converges in Lr. h then follows that the Fourier series of ,9"o coincides
with that of /, and hence S- : f ,P-a.e.

Chapter 2

Value Distributions of GeneralE)irichlet Series
2.I Introduction
A general Dirichlet series is a series of the formfo^"-^^", s:o*i,teC,

(2.1)

m:l

where a- e C, and {)-} is a Dirichlet sequence,0{Ar()z("';.\--+oo.

Suppose that the series (2.1) converges absolutely for o ) oo and has the sum /(s). Then/(s) is an analytic function in the half-plane D:: {s € C: o;, oo}.
Limit theorems for general Dirichlet series on the complex plane, in the space of analytic

functions as well as in the space of meromorphic functions have been studied relatively
completely through papers [18, 19, 37,32,33, 34, 35]. Let us mention here the most recent
results. For ? > 0, denote by ,, the uniform probability measure on [0, T). Let IJ(D) be
the space of analytic functions on D equipped with the topology of uniform convergenceon compacta. Then the limit theorem for the absolutely convergent general Dirichlet series
in the space of analytic functions was proved in [31].

Theorem 2.1. There erists a probabi.li,ty rneasure P on (H(D),8(H(D))) such that thesequence of probability measuresu7(r: f (t+ir) e A), Ae B(H(D)),.

conuerges weakly to P as 7 -+ oo.

Suppose that /(s) is meromorphically continuable to a wider half-plane Ds ;: {s €C: o ) o0j,o0 ( o*. Moreover) we require that all poles of /(s) in Dg are included in acompact set and that the following two conditions are satisfied.

(i) /(") is of finite order in any half-plane o )- o1(o1 ) o0), that is, there exist constantsrz) 0 and to ) 0 such that the estimate+ it) : o(lll")'holds uniformry for o , or'.(oltl ) to'(2'2)

Chapter 2. General Dirichlet Series

34(ii) For o ) os such that {o -li,t: I e R.} does not contain any pole of /(s),Tノ(σ 十 )2ごt=ο (T), T→ ∞・

(2.3)

Jι

C U {oo}, and d be the sphere metric on C* deined

Let C。 。 be the Riemann sphereけ
α(Sl,S2)=

21sl― s21//1+sl12ν //1+s22'に 銅 =濤

蔦

嗣 =頃 甲

ν…

QThis metric is compatible with the topology of C-. Let M (D6) denote the space of meromorphic functions g: Do + (C-,d) equipped with the topology of uniform convergenceon compacta. Then the limit theorem in the space of meromorphic functions was obtainedin [32].
Theorem 2.2. Suppose that condtti,ons (2.2) and (2.3) are satisfied. Then there erists a
probabi,li,ty measure P on (M(Do),8(M(Ds))) such that the sequence of probabi,li,ty meaSUTCS

u7(r: f(s+ir) e A), AeB(M(Ds)),conuerges weakly to P as T -+ x:.

The limit theorem on the complex plane was obtained in [33].

Theorem 2.3. Suppose that condi,ti,ons (2.2) and (2.3) are sati,sfied. Then for each o )os, there erists a probabi,li.ty measure Po on (C,6(A)) such that the sequence of probabili,tgrneasures

u7(r: f (o+ir) e A), Ae B(C),conuerges weakly to Po as 7 -+ oo. In other words, for each o ) os, the li,mi,t di,stri,buti,onof f (o't i,t) eri.sts.

To identify the limit probability measures in the above three theorems, some additional
conditions are necessary. Suppose that the sequence of exponents {)-} is Q-linearly
independent. Let Q: flff:, 1-be the infinite-dimensional torus as defined in Section 1.3
and ms be the normalized Haar measure on f,). Assume further that, for o ) os,

Σ

lα

π 12θ

‑2入 れσ(10g m)2<∞

.

(2.4)

m=1

Then it was proved in[34]that for σ >σ O,the series
OO

σ
λ
η
mω (m)θ
∫し,0=Σ αm=1converges almost everywhere, and hence is a complex-valued random variable on the prob-

ability space (fl,B(O),mg). Moreover, the limit probability meastre Po in Theorem 2.3
coincides with the distribution of the random variable f (o,r). In addition, under conditions (2.2)-(2.4), it was proved in 119] that /(s,0.,) defined by

λ
ms
ω
れ
ノ
(m)C
い,0=Σ αm=1General theory

is an ff(Ds)-valued random element and the limit probability measure in Theorem 2.2
coincides with the distribution of f (s,u).
This chapter is devoted to identify the limit probability measures without assumptionof linear independence of {)-}. Under conditions (2.2) and (2.3), we will show that for
fixed o t o11, f (o *zf) is a B2-almost periodic function with the Fourier seriesf (o +ir) - i

a-e-^^or--i\-t.

m:)-

Therefore, the limit distribution of f (o *it) exists and coincides with the distribution ofan L2(Q,P) functionmf (".r)::a^e ^^ow(m).

Here (Q,6(CI),P) is the probability space developed in Section 1.3. Moreover, as weproved in Chapter 1, the series f (o,r) converges P-almost everywhere without any further
assumption. Consequently,

f (".r):ia*w(m)e-^^'m:l
is a well-defined .F1(Ds)-valued random element on the probability space (Q,6(Cl), P), andits distribution coincides with the limit probability measure in Theorem 2.2.

2.2 General theory
The main aim of this section is to approximate the function /(") bV a sequence of absolutely
convergent Dirichlet series. If the function /(s) is analytic in Ds, we can find this kindof result in 135, 18]. We begin with a result on the mean value of absolutely convergent

Dirichlet series.

Theorem 2. (cf. [45, S9.5J). For any oy ] oo, uni,formly tn o 2 or, we haue
lfToo

lil
_ltt" + it)fdt: t la^l2p-2^^o.
t --+x) #Zl J| -T
m:I
The following formula is known as Perron's formula . We will use the Dirichlet series
defined in that formula to approximate the function /(s).

Lemma 2.5 (ct. 145, $9.43]). For 6 > 0, ) ) 0, and c ) 0, c ) oa - o, u:e haue:, 1,i ['*'*f (+)* w)5-*dw.
fL*rtl"o-"-^- s"-k^nd)^"\^/" f@2ri)J"_;,o

(2.5)

n-f

where I denotes the Gamrna functi,on.

Let- /^\ t:g.l.a(s)S--.\-s--(e^-6)^( -> 0,d > 0).
" '' . ,\L.a-e-o^-p

m:I

It is clear that the Dirichlet series gt,5(s) is absolutely convergent for any s € C. The
sequence {gr,a(")}o approximates the function /(s) in the following sense.

Chapter 2. General Dirichlet Series

36

Corollary 2.6. Let K be a compact subset r,n D. Then for fired l > 0,/t sek l/(s7-*- I Js

Iim lim sup +
5-+O

,,ro

+ it) - ss,5(s + it)ldt : 0.

Proof . Let L be a simple closed contour lying in D and enclosing the set K and let d6
denote the distance of L from the set K. It follows from Cauchy's integral formula thatsup l/(s +it) -e.\,6(s+ir)l| '"./v^.o\" | (w,/l \<s€K:f lf(z\' +' it\ - ss,5(z + i,t)lld,zl,
2t61a JTtruu)then by the Cauchy-Schwarz inequality,

/

\2 .

(sup l/(s + it) -e),6(s * Ol)

tLt r -t it)- s^,6(2 + iqf@zl.
e#ip J"lttz

Here lll denotes the length of the contour.L. Thus when 7 ) max"q;lImzl,

/7 fr
.-\'
(; /. su; l/(s + tt) - e).6(s + it)ldt )7 frl.. + it).\2 dtrt)l)= ; J, [sup- e),d(s +l/ts.+ l, (#* l,ve+ tr) - esa(z+ '1l2laz1) at:#tr I,G l,' v'z-t it) -exa(z 1-it)l'at) la'lz-r it)l'ar)* it)'/ ta.ll"Gl_i,,r,".
#;p"
1 f" ,, + it). 4lLl2,"- sxa@ + it)f il*
d#ip:yy,
J_,,1f"ze.,.a(Rewhere o1 : Irlirtz€ lP;e z > oo. Now, uniformly in o ) oy,"

oo

1 f21-

1*. - J-rrltl"+ it) - s^,6(o + iill2: t

m:7

la,nl2s-2\^o (1 - e-("^-d)r;2,

by applying Theorem 2.4 to the function /(s) - g.r,a("). Therefore,

/t rr

\2

Iimsup ( ; / supl/(s+ 1f) -g.r.a(s+tf)ldt )T +a \r JO seK/

4tLt2 3

< #,Im:I lo*l'"-zx^ot (l - "-(e^^

6)^

Sz .

The above series is dominated bv lil:, laTnl2s-2\^or < oo for any d > 0, and each term
converges to 0 as d -+ 0. Thus by the dominated convergence theorem, we arrive at

T翌
l撃

F:ズ

胆鶏
The proof is complete.

=潔

騰 +0州

S+0隣 =0□″
００General theory

If there is no pole in De or /(s) is analytic in D6, we have the following version ofTheorem 2.4.

Theorem 2.7 (ct. 145, S9.51]). Let f (s) denote the analyti,c conti.nuat'ion of the functi,onf (t),o ) oo to the half-plane o ) a. Assume that f (s) 'is regular and of fini,te order foro ) o, and that

洗十=OC),T→ ∞
・
い
の
「
￡げ
場Ъ
も
いの
レ=2♭ 硼
鳳寡￡げ

rん θη

p.6)

(2.7)

ηα
ηνstrゎ ct<σ lく σくσαttη グθrmι νづθr σ>α ,α ηメ2・Consequently if∫ (S)iS analytic in DO,the statement of Corollary 2.6 is still true forany compact subsetス「 of」 90. We are no、 v in a position to extend Corollary 2.6 to ourconsidering case in which all poles ofノ (s)in DO are included in a compact set.It thenf01lo、 vsthat the number of poles are flnite. The poles and their orders are denoted by

Sl,… ・,Sr and ηl,.… ,η r,reSpectively.

Proposition 2.8.Zθ ι](bθ a cθ 衛フαcι Sttbsct tη DO.Tん εη ル r fχ θα λ >σ α― σO+1,

鳳

υんctt ιO ts αpο

sづ

1響 lイ

K針 0‑嫌

避
JQゎ づ
η
ι
づ
υ
θttα J ηanι cr sα ιθ腱勢刊 隣 =Q{Im S}十 ιO>max{Im sl,・ ・・,Im Sr).

Pη 9二 From COrollary 2.6,we can assume without loss of generality that the compact
subset κ is included in the strip σO<σ <σ α―
卜1. Let Z be a silnple closed contour lying―
in the strip σO<σ <σ α 卜1,enclosing the set K and
聖 ]{Im S}十ιO>max{Im sl,… ・,Im Sr)・

Then Z lies in the strip σlく σ く σ2,Whereσ2=理 TRes<σ α+1.
l=mttRes>σ O, σ
Choose α ∈ (σ o,σ l)Such that all poles sl,… ・,Sr lie in the haliplane σ >αFor s=σ +tt with σ∈[σ l,σ 21 and S￠ {Sl,… 。
,ST},by mOVing cin the formula(2.5)to.

C=α ―σ,we pass a pole at ttl=0,with residue λ,ttl=Sγ ―S.
∫(s),p01eS at trl=sl― s,… 。
Since λ>σ ―α,no other pole is passed. Therefore,by the residue theorern,v″ e obtain

S+0お ω
αυズ
ぬ0‑胴 =万 鳥。
頂/11[II・ や十ノiゞ LRes(「 (11)ノ (S+復 )δ

tt,SJ―― S)′=1=:」 (S)十 J(S).

(2.8)

Observe that

Resl「 lや 炸 十ぽ

勁→

=菖 Ψ

鰊 姜声 う°L=Chapter 2. General Dirichlet Series

38

where (k) denotes the k-th derivative with respect to ur anda(k. s,) =lim l(t,, * w)(w -(""r - s))n,)'n'-o "(,,ll,--]lrti- k - *l)! u'Ji,^-'L\"I.

Thus, for fixed 6 ) 0,
\

/n

-s,|
J(s): o(r,lrrtlls; A
n:max{nr,"',n,}-r'
1,
t/
\ffi'

(2'9)

Now, an argument similar to the one used in the proof of Corollary 2.6 shows thatsup l/(s + it) -g.r,a(s + tt)lat\e
f sek'\7 Jt.
/: o ( ?'p #[" ,rro + it) - s^.6@+ irtl'at)ar Jtt\ae[a1 .a2].

/

where f1 : min"Ez,{Ims}*to ) max{Imsl,...,Imsr}. For o € [or,o2] and t)h,thepoint s : o -l it does not belong to the set {s1,...,s"}, thus the relation (2.8) implies
lsx.d@+ ?r) - f(o +ir)12 < z(VO + iill2 +Q@+ir)|'?).
Note that in the proof of Theorem 2.7 (see [45, S9.51]), we have the following

1fT; J-rll(o + iDl2dt : 0(62"-2")
uniformly with respect to 7 and o e lor, o2]. lt follows that
Iim limsup sup+ f' [@ + iDl2dt

d-+0 7e6s- oeloi.ozl 41 J tr( d-+0lim limsup sup +41 f'
7-_y- oeloi,o2)[@ + iDl2dt: d-+0
lir4 o(52o'_2oS : g.

J 2,f 'On the other hand, by Stirling's formula, there is a constant A > 0 such that uniformly
in the strip o/ ( o { d", we havelf (o + it)l: O(e-Attt1, f -+ oo,
where o' < minT{Re((s, and o" ) maxi{Re((ri "t) l\}"") l\}
hand. This, together with (2.9), implies thatbeing chosen before-

1r2T/tr2T\:0.
Iimsup sup +l/
lJ@+it)l2dt:o(limsup--1'
T+C oqlo1.ozl 41 1,,\ r-o' 4T Jr, "-Atdt)/
The proof is complete by combining the above two estimates.
As a consequence of Proposition 2.8, we have the following.

Proposition 2.9. Let K be a compact subset i,n Ds. Then for fired ), ) oo - o6 * 1
lim Iim sup.,ro d(/(s + i.t), ss,5(s + it))dt : 0.
/" se6
d+0 746e- :I ../6n

2.3. Limit theorems for general Dirichlet series on the complex plane

2.3 Limit theorems for general Dirichlet series on the complex plane
Let (O, B(A), P) be the probability space defined in Section 1.3. We consider Bfo, to > 0
being fixed, the space of one-sided Besicovitch almost periodic functions, that is, functions/: [0,oo) -+ Cu {oo} which belong to the linear closure of {ei^t:,\ e R} with respect tothe one-sided Besicovitch q-norm (1 ( q ( -),ll/lln,,, :: limsup

/t 7T

\l/s

(; l" lf (t)lqdt)

Then the following results for one-side Besicovitch almost periodic functions are similar
to those of Besicovitch almost periodic functions. They are taken from Theorem 1.8,
Lemma 1.9, Theorem 2.10 and Theorem 1.23. Note that in this chapter z7 denotes theuniform probability measure on [0,7] while in the previous chapter, z7 denotes the uniform
probability measure on [-?, ?].

Theorem 2.10. (i) The sequence of probability measures {Q7} conuerges weakly to P
as T -+ @, where

Qr(A) :: ur(r : (e-i'\^1 ex € 1) , A € B(CI).

(ii) {c"'(m)}meN ?s an orthonormal system in L2(Q,P).
(iii) Let /: [0, oo) -+ C be a one-si,ded B?^-o.p. functzon wi,th Fou,ries series of the formoof(t)-la^e-i),,,t.
M:I

Th,en,oof(u):\a*a(m)m:lconuerges forP-a.e. aJ € CI and conuerges in L2(Q,P). Moreouer, the sequence of

probabi,lity Tneasuresu7(r:f(r)eA),

AeB(C),

conuerges weakly to the dtstributi,on of f (u).

Lemma 2.LL. For o ) os, we haue f (o + it) € B?^ wi.th the Fourier seriesf (o +ir) - i

eme-^^os-i\*t,tn:lwhereto2 0 'is a number such that {o -ti,t:t) ts} does not conta,in ang pole of f (s). In

れσ
洗
も
=2降 州
けの
「
鳳lイ げ

part'icular.

<∞(2.10).

Pη ρ′ Fix λ>σ α―σ。+1.For each η∈N,we deinegη O):=gぇeλ.15・ l=ΣαmeXp{一
E
272=l

び

λm

λ→ λ

}c
m=1

λnS=Σ

α mυ (鶴 ,→ C入れ S.

Chapter 2. General Dirichlet Series

40

The Dirichlet series gn is absolutely convergent for s∈ C.Thus,it is clear that gη (σ +jι )∈Bλ br any to≫ 0,whCh has the bnowing FOuHer seHesυれηθい,→ C λ mσ
し+0〜 Σ α

cttλ

帯
.

222=1

For σ>σ O,let tO>O be a nulnber such that{σ tt tt:ι >to}∩ {Sl,… .,Sr}=0・ Then inκ ={σ },We have宙ew of the proof of Proposition 2.8 with鳳1摯一頸 σ切 陶 =Q

σ

LttFlイ Ⅸ 刊

Note that for each m∈ N,υ い,η )→ O aS η→Oo.Thusノ (σ tt jι )∈ Bλ and

一
似
λσ
θ
α
mc―
ノ +tt)〜 Σ〕
OO

m

mι

(σ

772=l

□

is the Fourier series ofノ (σ tt jι )・As a consequence of Theorell■

2.10 and LelnIIla 2.11,、 ve have the follo、 ving。

Theorem 2.12. Fο r σ >σ O,

λσい
m「 ηω
ノ
し,0=Σ α)π

=1

Sル rP― α・C.ω ∈ Ω αηd θθηυθη cs jη L2(Ω ,P)・Cθ ηυCη θbヴ

ν θttθ υcら ιんθSθ gacη ccげ Pη bα ―ιじιν mθ αsarcs
r/T(γ :∫ (σ 十を7)∈ ■),ス ∈β(C)Cθθηげ ∫(σ ,ω )α S T→ ∞・ιαんοιんθα,strづ bttι じηυCη CS υθνι2.4

Lillllit theorems for general E)irichlet series in functional

spaces2.4.l

Absolutely convergent case

Recall that D=(s∈ C:σ >σ α}and〃 (D)denOtes the space of analytic functions on DI),ω ∈ Ω,letequipped、 vith the topology of unifornl convergence on compacta. For s∈f (t,r),: D a*e-^^'w(m).
Then /(s, c,.') is an fl(D)-valued random element on the probability space (Q, B(O), P)' In
addition, we will prove that f (t,r) is continuous as a mapping from f) fo H.(D).Indeed,let {u@)7 be a sequence converging to r,,r in 0. We need to prove that {/(s, ,("1)} convergesto /(s, w) in H(D). Given a compact subset K c D,let o1 : min"671Res ) ao. We have:』λmゆ 0い )ω いαノ。°
) 炸 ,0く
Σ
mlθ

'ω

1

)・

2.4. Limit theorems for GDS in functional spaces
Since !ff:

41

llarnle-^-"t ( oo, it follows from the dominated convergence theorem that

;3; l/(s,a\)) - /(s,r..')l -+ o as n -+ @.

Thus, the mapping
is continuous. Consequently, the sequence of probability mea"f(s,r,.,)sures {Q7l(",r)-t}r on (II(D), B(H(D))) converges weakly to P/(s,r,,')-1 as ? -+ oo.
Obviously, we haveu7(r: /(r+ ir) e A): Qr(r: f (s,u) e A), Ae B(H@)).
Therefore, we have just proved the following theorem.

Theorem 2.I3. The sequence of probabi,lity measuresu7(r: /(r+ ir) e A), AeB(H(D)),
conuer-ges weakly toPy asT -+ x, wherePl denotes the di,stri,buti,on of the H(D)-ualued
random element f (",r).

2.4.2 General case
Recall that Ds: {s € C: o ) os}. There is a sequence {Kn} of compact subsets of Ds
such that (i) to : ULr I("; (ii) Kn C Kn+Lj (iii) if 1{ is a compact set and K C Do,
then 1( C Knfor some n. Then for f ,g e H(Ds), Iet

2η1+Sups∈
ガ =Ё
耳
ギ黎響
写饗鼎
蛇
名
,の

￨ノ

(s)一 g(S)￨・The topological space∬ (DO)beCOmes a complete separable metric space.Similarly,for

ノ,ク ∈χ (Do),let

ノ:=Σ rnI SupS晩κπαo,gO)
ズ
<ノ

,の

+sups∈(∫ (S),9(S))・

Then if(DO)alSO becomes a separable metric space.
Lenll■ a2.14.

ノ(S,ω )=Σ αmC

λがω(m), S∈ D0

m=1

―
づ
s αTcι ιααιacα ttη αθm cι θ
ttθ ηt.Bcsjα cs,ヵ r αηνf"θ αS∈ DO,
げηCα ∬(Do)― υ
mSω
mc λ
∫
←,0=Σ α
い
)

π

6θ

=1

ηυ
Cη CSル rP― α.C.ω ∈Ω αηごθθηυθηcs tη Z2(Ω ,P).

P知 のl By TheOrem 2.12,for any σl>σ O,the series

σい
η
∫
mc λ
けbO=Σ α
lω

)

772=1

converges for P― a.e. ω ∈ Ω. Therefore by a fundamental property of general Dirichlet
series,for P― a.e.ω ∈ Ω,the series
mSω
C λ
∫
η
C,0=Σ αい)m=1Chapter 2. General Dirichlet Series

42

converges uniformly on each compact subset of the half-plane {s e C : o ) oy}. Let
A2 denote the set of a,r € f) for which the series f (t,r) converges uniformly on compact
subsets of the half-plane {s e C ; o ) os+ 7ln}. Obviously, P(A") : 1 for all n € N.
Now if we take
m

A: )

A,.

n:7

then P(,4) : 1, and, for c.,,r €,4, the series /(s,r,,,) converges uniformly on compact subsets
of. Ds. It follows that /(s, c..,) is an fI(D6)-valued random element defined on the probability
space (Q,6(0), P).
tr
For each n € N, we define a random element gr(s,a): Q -+ fI(Ds) as

9n(s,

w),:

a^u(m, n)e-^^" u(m).

:Lemma 2.15.

J$nrer lnb,(.,r),f (.,r))']:0.

ProoJ'. Let 1( be a compact subset in D6. We will show that,,$e(el llh'(w)12):g'(2'11)

where

h"(r),::;rp lg"(",r) - /(',r)1.
To prove (2.Il),let -L be a simple closed contour lying in D6 and enclosing the set 1{ andIet d denote the distance of .L from the set K. For c,.r € f) for which f (t,r) e H(Do),Cauchy's integral formula implies thath.(r):::p lgn(r,r)- f(r,rll < roa!

l"tn,(",r)- f(2,w)lld,zl,then by the Cauchy-Schwarz inequality, we obtain^lh,(r)1'? E

t/.1 f--r:!
I lg,(z.u) - f(z.r)l'larl.
\.z7To)" J L

Let o1 :rnirr"67,Rez ) o6. For z: o *i,t(o 2 ot), from Lemma2.l4, we haveη ぃ),lin z2(Ω ,Pl),C λれル,0=Σ α
zω

η

=1

、
vhich implies

EIPllgη2c2λ mση
し,0‑ル ,01句 =Σ α′11‑υ (m,→
れ

=1

くΣ
m=1

αm21̲υ ぃ ,2)12c2λ mσ lく ∞.

43

2.4. Limit theorems for GDS in functional spaces
Therefore,

E(P)flh,(o)l2l <tr"rf ,, u)\ - f (z' w)f\'l)I
f-lrl
L\## Jrln"(z':,=1tr1.- [ ntPtlg,(r,w)
LrY'r\-'- / - lf(z,u)l2llazll2t6)2 J 2oolrl2< 'L' I
en6p

lo-l'll - u(m,n)l2e 2^^ot.

^,_,

Our desired result (2.1I) follows from the dominated convergence theorem. The above
result holds for any compact subset K in Ds. Therefore, taking the definition of themetric p into account, we obtain,$ o(el IPG,( a) f (' ,') )'] : o',,r

The proof is complete.

Corollary 2.16. (i) For any 6 ) 0,/\
iim P lob,(. u). f (..,t)) >-, ) : o.
(IU

Pn. -!+Py

as n -+ @)(2.12)

Proof. (i) follows from Lemma 2.75 by Chebyshev's inequality. (ii) follows from (i)Theorem 0.6.

ｙｂ □wherePn. andPy denote the di,stri,buti,ons of the H(Ds)-ualued or M(Ds)-ualued random
elements g, and f , respecti,uely.

For every compact subset K of D0, Proposition 2.9 claims that

J51

t,L* + I,' ::p d(e,(s + i,t), f (s + it))d,t : 0.

Thus, by Chebyshev's inequality, for any e ) 0,-$li-supur(,\ I p(s,(. * i,r), f(. + tr))> r) : o.
n--+oo ?_+oo(2.r3)

Since the Dirichlet series grr(s) is absolutely convergent in D6, it follows from Theorem2.73

that(2.14))二 》P,.as T→ ∞where the weak convergence still holds in the space of meromorphic functions M(D6).
Therefore, (2.I2) (2.14) imply the limit theorem for /(s) in the space of meromorphic
1/T(γ :gη (S+づ 7)∈ ・,functions M (Do).

Theorem 2.L7. Suppose that condi,ti,ons (2.2) and (2.3) are satisfied. Thenf(r,r): ia^e-^^"w(m), s € D6

rn:li,s a well-defined H(D6)-ualued random element. Moreouer, the sequence of probabi,li,tyrneasuTes

u7(r: /("+ ir) e A), Ae B(M(Ds)),conuerges weakly toPy asT -+ n, wherePy denotes the di,stri,butzon of f (t,r).

Chapter 2. General Dirichlet Series

44

Remark 2.18. If thestripDl: {s € C:o1 <o <oz}(oo < 01 < oz ( -) containsno
pole of /(s), then in view of the proof of Theorem 2.17 we can assert the following.
"The sequence of probab'ili:ty measuresu7(r: /(s+ ir) e A), Ae B(H(D1)),
conaerges weakly toP y as T -+ n, whereP y denotes the di,stributi,on of the H(D1)-ualued

random element f (t,r)."Remark 2.19. Theorem 2.72 and Theorem 2.I7 are extensions of the main results in 134]and [19], respectively. Comparing with proofs in [34, 19], the basic idea does not change
but a number of arguments are reduced. For instance, we use .L2-convergence instead ofusing the tightness of measures and ergodic theory.

Part II

Besicovitch Limit-periodic
Arithmetical F\rntions

45

Chapter 3

Limit-periodic Arithmetical
Functions and The Ring of Finite
Integral Adeles
3.1 Introduction
Chapter 3 and Chapter 4 mainly concern with arithmetical f'unctions, real or complex
valued functions defined on the set of all natural numbers N. For simplicity, we sometimesヽヽ︑ ︲ ′ ノ

η

∫

′ ︲ ヽ

Ⅳ▼
ん祠

／

ノ￨19
‖

１一
Ⅳ

中
﹄
・

omit the term "arithmetical".
An arithmetical function f : N -+ C is called limit-periodic if it is a limit of periodic
arithmetical functions under some Besicovitch q-(semi)norm (1 ( g < oo),

Limit-periodic arithmetical functions, in some sense, look like random variables because
they possess mean values and limit distributions. Here the limit distribution of an arithmetical function .f is considered as follows; if the limitrt(/l,u-^lit *,n I""on:lsRe/(n) + JttIml(n)). (s.r) e R2.

exists and it coincides with the characteristic function of some orobabilitv distribution on
R' - C, then we call it the limit distribution of /.
This chapter deals with the problem of finding an appropriate probability space where
Iimit-periodic functions can be considered as random variables. The ring of finite integral
adeles 2, bgether with the Borel field B(2) and the normalized Haar measure ), is shown
to be a good candidate. Indeed, let Dq denote the space of q-limit-periodic functions.

Then every function in Oq can be extended to a random variable in Lq(Z,)). The limit
distribution of the original function coincides with the distribution of the extended random variable. In addition, the quotient space Dq of.^Dq with respect to^the null-space
N(O01 ,: {f €Dq: ll/lln:0} is isomorphic to Ls(Z,^), which means (Z,B(Z),)) is agood candidate. In fact, the ring of finite integral adeles 2, which was initiated by [39]and has been studied by several papers and books [10,28, 36,37,44], has become an
useful tool for studying probabilistic properties of limit-periodic functions. It is called thelimit-periodic compactifrcation of Z. Besides, to investigate the probabilistic properties

Chapter 3. Limit-periodic Arithmetical Functions48

of arithmetical functions, there are other compactifications, for example, the almost evencompactification used to investigate almost even functions 127]; the Bohr-compactification
used to investigate almost periodic functions [37] and the Stone-eech compactification
used to investigate a wilder class of arithmetical functions which contains the Mobius
function 125].
This chapter also concerns with the convergence of Fourier expansions of limit-periodic
functions. First of all, we define the scalar product (f ,S) of functions ,f,grN -+ C to be
the limit\f .g) ,:,Jg:

r

-f,'

i n:If rt,)e(,)

provided that it exists. Then {eo}oeqlzbecomes an orthonormal system, where eo stands
for the function edi n t+ exp(2trfian). Note that the space of limit periodic functions isspanned by {eo}oee1z. Moreover, for each f e Dq, the Fourier coefficients {(f ,e.)}oeqlz
exist and the formal series,^H,urf.eo)eo
is called the Fourier series of /. Next, for each n € N, we consider a finite Fourier expansion

ofafunctionfeDqas

島 (ノ ):=(ノ ,Cα /r)Cα /r,,.,.ili;,,,:,
where (a, r) denotes the greatest common divisor gcd(a, r) of a and r. Then, we have thefollowing result about the convergence of Fourier expansions of limit-periodic functions

税 (∫ )一 ∫￨19→ O as η→ Oin Z, ノ∈Ds.
‖
From this a similar result for almost-even arithmetical functions easily follows. These
results give an apprbximation for limit-periodic functions (resp. almost-even functions) byperiodic functions (resp. even functions), and they are the generalizations of 143, Theorem
VI.5.11.

The convergence of Fourier expansions in the special case q : 2 is easily seen since{e.-}d€elz is an orthonormal basis of the Hitbert space D2. To extend this result togeneral q, the idea here is to apply the interpolation of norms and of linear operators
tool, a major tool in harmonic analysis, which was successfully used by Bochner to provethe convergence of Fourier series on Ln(ll,2rl),I < q < oo (see [26, Chapter IV]). The
isomorphism between Dq and. 1u12,\) makes it easy to apply this tool as we will see inSection 3.3.

This chapter is organized as follows. In Section3.2, the probability space (V,,BjV,),^)
is introduced. Then the natural isometric isomorphism 7n between Dq and Lu(Z,),) isdefined. Moreover, the inverse of Tn can be obtained by Lebesgue's density theorem(Theorem 3.21).. {he beginning of Section 3.3 deals with the convergence of Fourier
expansions in Lt(2,)). By using the interpolation of norms and of linear operators tooland the dual property, we obtain the convergence of Fourier expansions in Lq(Z,,\) for
all 1 ( g < oo. At the end of this section, we deduce the above convergence in Dq(Theorem 3.27) by the isometric property of To and its consequence on the convergence ofRamanujan expansions of almost-even functions. The natural extensions of additive andmultiplicative functions will be considered in Section 3.4.

3.2. The ring of finite integral adeles

3.2

3。 2。

The ring of flnite integral adeles: basic properties andconnection to lilnit… periodic arithmetical functionsl

The ring of inite integral adeles and sOme basic properties

This section deals、 Ⅳith the construction and properties of the ring of lnite integral adeles.

Results are taken iom p9,4到 .For a prime p,the p― adic metricら is deined byαp(χ ,ν ):=inf{p J:pJI("― ン)},・ ,ν ∈Z,
where p̀￨(χ ―ν)means that(χ ―ν)iS diViSible byノ .The completion of Z by α
P is denotedby Zp. By extending the algebraic operations̀1'and̀×' in Z continuously to those in

ZP,the compact metric space(ZP,ら )becOmes a ring,called the ring of padic integers.
In particular,(ZP,ら )iS a compact abelian group with respect tò+'.According to thegeneral theory of connpact groups, there is a unique normalized Haar measure ttp｀
Vithrespect tò+'on the lneasurable space(ZP,β (Zρ ))・
Deinition 3。 1。

(i)Let{P,}窪 1,2=Pl<p2<・ …,be the sequence of all primes.

(ii)Put

2:=Π ら,,.λ :=Π λ
2.
t=l

j=1

For"=(″ o),ν =(ν ,)∈ Z,We deine‐し 協,0‐ SL多 ら ら
ρ
し
の,χ ν
),χ ν
=Oj銑
,十

tt

tけ

)・

t=1

By these deinitions,Z becomes a ring,calledι
んcれ ηθげ fη づιθれιιααcι es.(Z,′ )iS"鶴again a compact metric space,and both̀十 'and̀× 'are continuous.In
particular,(Z,ρ )is a compact abelian group、 vith respect tò1'and itS normalized Haar lneasure on the
Borel ield β(Z)is nOthing but入.

Deflnition 3.2。

(i)ヽ Ve identitt Z With the diagonal set{(2,η,。

…}⊂ Z.
・・)∈ Z× Z× ・0糧

鯰 に うT糖 (脱 夕革ぢ,格 ∬ i∬ 器 iイ ir(二 靴 配 ご )メ ょ
fOr∬ ∈Z and N∋ 鶴 ≫ 2,there exists a unique ι∈(0,1,… ・,鶴 ‑1}SuCh thatχ― ι∈ ?η Z.This J is denOted by″ mod m.For m=1,we always set χ mod m:=0.
Obviousl)L if″ ∈ Z,this deinition coincides、 vith the usual lnodulo operation.

(iii)For χ,ν ∈ Z,、 ve deine the greatest common di宙 sor of χ and ν bygCd(χ ,ν ):=Sup{m∈N:(″ mOd m)=(ν mod m)=o}・

Obviousl)Ъ for χ,ν ∈ Z,this deinition coincides、vith the usual gcd.

Let us give some fundamental properties of Z andλ.

■SSamc ml,… .,m̀∈ N
Lemma3.3(ChineSe remainder theorem,cf.21,Theorem 1211)。
―んαιη =
ιθ bc cθ p7・ 2mc.Tん θη,ル r αην αl,・ ・・,α ι ∈ Z,ι んcた c"づ sts αη η ∈ Z sacん ι
js ηtsし η
ι
θ
mod
。
.助
り
=1,…
c(mOd ml),を
9zc T
Π:=lm・,ι

Lemma 3.4.N′ :={(η ,2,… ・θη
SCれ Z・)∈ Z:η ∈N}づ s α.

Chapter 3. Limit-periodic Arithmetical Functions

P"のl The chinese remainder theorem implies that fOr any ι,m∈ N and any ηl,… .,η ι∈
N,there exists an η∈N such that n=η づmod pr,づ =1,… .,ι .This means that N′ isvith respect to the metric ρ.
dense in Z× Z× …・、□={(η ,2,… ・)∈ Z:η ∈ Z}(Deinition 3.2(1)と by a bij"tion→Z′
Z∋ η (2,η ,・ …)∈ ,Lemma 3.4 implies that Z is a dense subring of Z.Thus Z is acompactincation of Z. It is called the lilnit― periodic compactiication of Z or N.
As we identitt Z With Z′

Lemma 3.5。

S CJθ scα αηαttθ η(i)Zθ ιp bθ αPrtttθ αηdブ ∈ N.T71enノ Zρ じ.

θごJsれ ηcι Pれ mcs αηαノ cN・(ii)五 θιP,9 ι

Tん θη υθんαυθノ

Z9=Z9.

Pη イ (i)It iS easy to see thatノ Zp={"∈ Zp:ら (″ ,0)く p′ },and hence it is closed.
Sinceら (χ ,0)∈ {pα :α =0,1,...,∞ }for att χ∈ら,記 mtt Wnteノ ZP={χ ∈ZP:+1},WhiCh implies it is open.
ら(・ ,0)<p ′(ii)ノ Z9⊂ Z9is obvious,so let us prove p7Z9⊃ Z9.To this end,it is enough to showthat there is an χ ∈ Z9such thatノ∈ N,there exists an"π c N such"=1.For any鶴that rmノ =l mOd 9772.Then for any η>777,w℃ haて (″ η―″πι)ノ =O mod9れ .Sinceχπ―χ772=O mOd 9772,which means that{χ 222}需=l iS a cauchy□
sequence in Z9.Then putting":=limm→ ∞ ,We haveノ ″=l in z9.

gCd(ノ ,9m)=1,We see"れLemma 3.6.Zcι 鶴∈N αηαι∈{0,1,一 ・,m‑1}.
(i)T72C SCι (鶴 Z ttι )'S Cι οSCα αηα θPθ η

.

O
(面

ρ

m:2→ p,4ぉ

=Qω い ρ={:観麻∬ttη ttθm。

,W加れιZη じ
寛
cん じ
sα ご
づ
θ
巧θ
)2=∪ 鷹51(m2+ι ),υ んづ.

P印 げ (i)Let n=Π ppα P(772)be the Lctorization of m into primes,where αp(m)=0except for initely llnany prilnes p. Then,Lelnlna 3.5 imphes thatm2=Π 鶴ろ=Π
p

pα

p

'い

)ら
,

and that each Pα p(m)ZP is closed and open. Therefore,γ ttZ is also closed and open in Z.

Finally,since the shift Z∋ χ 吟 (χ tt ι)∈ Z iS a homeomorphism,mZ tt ιis also closed

and open.
1({■(ii)Since(i)implies that ρ尻 )=m2is closed and open,the statement is obvious.
(iii)From the denseness of Z in Z,and tom the COntinuity and closedness ofthe mappingχ tt mχ 十 :,it fo1lows that ttZ tt

ι=鶴 Z tt ιo Since Z=∪ 鷹 51(mZ ttι ),thiS implies2=∪ い2+ι

).

̀=0

′

Next we check the dittointness of this union.Let鶴 ≫2 and ι ∈(0,… ・,m‑1}bedistinct integers.By o,ス :=い 2+ι )∩ 鰤2+J′ )iS Open.Ifス ≠0,then Z∩ ス≠0,4,we see fronl the observation of
because Z is dense in Z. But then,taking an η∈Z∩ ノ
,ι

←
)that

Pい ),っ :p五 me.
2(m), ら(η ̲〆 ,0)く p αズη̲ι ,の くp α
朽′This implies that pα 2(m)￨(ι ̲J′ )fOr each prime p,that is,7721(ι ― ι),WhiCh is impossible.
□
Thusノ 4 should be empty.

κυThe ring of finite integral adeles

Corollary 3.7. For ang I e Z, the rnapping(l + r) mod rn

Z∋ ″ 吟∈ 10,1)i,s continuous.

Lemma 3.8. For any I e Z\ {0} and" ang A e 812), we haue tA e B(2) and1)(1,4): ;^(A).

(3.1)

l,l

Proof. Since 2 is a complete separable metric space and the map Z ∋ χ 卜〉 ιχ ∈ Z is
one-to-one and measurable, we have lA e B(2) (cf.[41,Theorem I.3.91).Let ν be a BOrel
probability measure on Z defined byλ(ι ■)

ν(■ )=■ ∈ β(Z).

λ 12)'(lιThen ν is clearly shitt inttriant,and henceν =λ ,so that λ(ι ■)=λ (ι Z)λ (■ )・

By Lemma

3.6 and the shift invariance ofハ ,we see

J‑1

1=λ ´
)=Σ ×12+の =lι l×
lι

lι

lの

,

づ
=0

from which,(3.1)immediately follows.

□

Lemma 3.9。
ιttsル η
焼し
θ
6ι じ
ο
η.動 θ
η{ノ (2)}η ∈ttη
∫:Z→ C be a cθ η
zづ S α(i)Zθ ιαづc sc9包 θ
ηθ
θ
んα
ιづ
s,
ル鶴 ι
νJづ 鶴じperづ θ
′ι

j―

;―

Vε >0,ヨ ιO, 7ぬ C IN SttCん ιんαιl∫lnodllprO)(2)一 ∫<ε , Vη ∈ Z。

(3.2)

(η

(ii) Conuersely, i,f {f(")}".u i,s^a^uni,formly li,mi,t-p^eri,odi,c sequence, then there ts a
un'ique continuous functton f : Z -+C such that f (n): f (n) for eachn €2.
Proof. (i) Obvious by the definition of the metric of 2.
(ii) If / is a periodic sequence with period rn € N, it is of the form f (") : DL, f (1,) p,n(, i'),n € Z. Then f1r) ,: DLr f (l)p^(" - i,),r e 2, i, th. continuous function with theproperty f lz : /. Note that a general / satisfying (3.2) is a uniformly convergent limit ofa sequence of periodic sequences, and hence it has again a continuous extension f Sin""
Z is densely embedded in 2, the uniqueness of / is obvious.
n
For a periodic sequence {S(n)}"ez with period m and its unique continuous extension

η

↓

σ
６
ｎ

ｗ

肺

初 ⅣＤ 向
０ ｌⅣ

飾
ｆル

χ
ガｕ

λ

嗣

′′厖
ｅ

ｅ
ｈ

ｈ

Ⅳ

In general, we

ｍ
﹄
い
ｉ
ヽ︲

j(r), it is easy to see that

(3.3)

Chapter 3. Limit-periodic Arithmetical Functions

52

Lemma 3.10。

J∫ :Z→

Cづ S Cθ ηι焼 鶴θas,ι んcηπZ
Xの =J吼 井
かの
苫 はれ∈
1人

.4)

The conuergertce 'is uni,form in ns € Z.

Proof. Let f tZ -+ C be continuous and set /r,o(r) : f (no* r),ns e 2. By the uniforrn
continuity of f , a family {fn,);no €Z) satisfies (3.2). For simplicity, set^.:lItftpTo.
Then we see for any ns e Z,

md→
ら
か 0× け れ 。
司く

0‑井 九
〇
Σ九
Σ

0儘 mod→

1井

CN.

ｒ

九

︹Ｆん 同

１ 一鶴

π
α

λ

鶴

ｄ
Ｏ

χ

ｍ

ん
ノル

By(3.3),

Ⅳ
∀

￨<ら

lk

+l‐

(島

鰤
十
0。
井菫 九

1)mOd

ヽ ノ

([全

胤

二
」
井≒
矛
卜 0+W
―
鳥
井
Σ原→
°

０

)=

九

(η l・ Od 7η

ｄ

れ
。

いΣ 同

￨ノ
l,1ケ
η=0

Σ︒
ハリ 消
ｒ

AlsO,by a silnple calculatiOn

ん。(r)

dれ

.

″=0

In the above and in what follows, the symbol ltl stands for the largest integer not exceeding

teIR.

From these, it follows thatlr,"(r mod m)Xdr) I lrf

ItL

lry:l mod.)lI c< #-llfll-.
fn:o,f,o(n"|"Therefore, choosing an "1y'6 € N so large that (ZlNs)mllfll- < e, we have that for anyl/ ) l/o and any n6 € N,lr., n6f.nr'-1

* n:o
I

I l.f t,lNa..t - rY

IJZ/(n)l< 3e.
|

il

I

Remark 3.11. As a matter of fact, (3.4) is a consequence of the following general theorem.

Theorem 3.L2. Let G be a compact group, and let r e G. Then, if the sequence {r"}7ri,s dense i,n C, i,t i,s uni.formly rl,i.stributed,; that,is, N-r Dlu:, 6rn cofi,uerles weakly to th,enormali,zed Haar measure of G os ly' -+ n, where 6r- denotes the Dirac measure at rn.

3.2. The ring of finite integral adeles
For details, see [30, Theorem IV.4.2]. In the present case, setting G ;: Z and r ::7,
we see the desired weak convergenceδ
→λas Ar→ ∞
れ
む
井ゞ
,

(3.つ

n:I

because N/, which is generated by r : 1, is dense in 2.

Theorem 3.L3. The shi,fts rr+ r*7 and,r+ r-

I onthe compactmetr'ic groupZ or"

ergodi,c.

Proof. The denseness of Zinh (Lemma 3.4) implies the ergodic properties of these shiftsI(see Theorem 0.16).

3.2.2 Connection to limit-periodic arithmetical functions
Let D be the space of periodic arithmetical functions, that is,

O :: Linc l"o/r rr : 1,2,....1 ( a ( r.gcd(a,r) : 1]'and let D" be the linear closure of O with respect to the uniform norm
sup l/(n)1, f , N -+ C.
ll/ll" ': n€N

Functions in D" are called uniformly limit-periodic functions or uniformly limit-periodic
sequences. Recall that the space of q-limit-periodic functions Oq is just the linear closureof O with respect to the Besicovitch q-norm.
Denote bv C(2) the space of continuous function" onZ endowed with the supremum
norm
ll/ll ,: sup lfrr)1, e c@).

f'x? I'

Then the spaces D" and C(2) are isomorphic under the isomorphism Tu:Du -+ C(2),which maps /^e D" 7o the unique continuous extension of / from N to Z. The preimage ofthe function f e C(Z) under 71, is just the restriction of / to N (Lemma 3.9). Let f e D"with / ,: T"(f) e C(Z). Then it follows from Lemma 3.10 thata
鳳井
二Kず =か が相and hence for any∫

∈つし,∫19=‖ rし (∫ )L9,
‖
、
vhere

(3.6)

功

ノ
にけ
nLぼ =an矧 気→ ,「 ∈

The continuous linear operator Tu on Du can now be continuously extended to acontinuous linear operator Tn: Dq -+ Lq(2,)) since O'is dense in Dq. For instance, let
f e D'|. Then there exists a sequence of periodic functions {/a}76 converging to f , that
it, ll"fr - fll, -+ 0 as k -+ oo. In particular, {f*}n is a Cauchy sequence in Dq. By (3.6),{f:"Uil}r, is also a Cauchy sequence in Lq(2,)), and hence the limit lim4** T"(fd exists

Chapter 3. Limit-periodic Arithmetical Functions
9(Z,入
in ι
).Let■ (∫ ):=limた →∞■(ル ).It iS Clear that
follows ti・ om(3.6)andfrOm the continuity of norms th就Ъ is Well deined;moreo(r itノ9=￨￨■ (ノ )L9, ∫∈D9.

(3.7)

This implies that Ъ is illiectiVe.on the Other hand,■ iS Suttecti(Since C(Z)iS dense
in Z9(Z,入 )and D9 is complete.Thus we ha√ e prOved the following theorem.

Theorem 3.14(p6,Theorem 2.司乃rlく 9<∞)。

,D9堅 Z9(Z,λ ).
Rcmα rた 3.1唆 (i)Let χ:2→ sl={ZCC:￨″ ￨=1}be a character of the compact
group Z,that is,χ is a continuous function satisfying
χ(″ +ν )=χ (χ )χ (ν ),χ,ν ∈ Z・

Then χ ==じ (θ α)fOr SOme α ∈ Q/Z.COnveFSely,for each α ∈ Q/Z,the image aン :=
写ん(Cα )iS a character of the compact group Z.Consequently,Theorem 3.14 implies
that 2Ъ (9)=Linc[亀 :α ∈ Q/Zl iS dense in Z9(Z,入 ).Functions in Linc[亀 :α ∈Q/Z]are periodic fllnctions.In addition,{亀 }α ∈Q/z iS an orthonormal basis of the:=のHilbert space Z2(2,λ )With respect to the scalar product(ス
(α ).

の

(ii)For a periodic fllnctionノ

)爾λ χ
スπ

,we alWttS think of■ (∫ )=乳 (ノ )althOugh場 (∫ )iS uniqueonly all■ ost surely

The following are some properties of 2b.

(i)■ iS Shift inttriant and multiplicati(ly shift invariant

τX∫ (η 十・))=■ (ノ )(2+・ ),(η ∈N),■(ノ (b))=■ (ノ )(b),(b∈ N),
whereノ (b)(η )=ノ (bη ),η ∈N and■ (∫ )(b)(χ )=■ (ノ )(b"),″ ∈Z・(ii)■(ハ =17b(∫ )・This follo、vs from the inequality(面 )If 1/γ

αl― bllく lα ― b,α ,bCC.

=1/p+1/9(γ ≫ 1)andノ ∈DP,g∈ D9,then Hё lder's inequality implies

ノク∈Dr and写(∫ θ)=場 (ノ )■ (g).

(iV)If∫ ∈D9,thenノ 19∈ Dl(143,Theorem VI.2.91)andTl(￨∫ 19)=￨■ (ノ )9・

P"げ げ (iV)・ WithOut loss of generality,assume that∫ ≫ 0.This proof is somewhat
similar to the proof of[43,Theorem VI.2.91.Let a:=Tl(ノ 9)1/9・

a=■

ヽ
ヽ
b Will show that

(∫ ).

9‑ん ‖
Given an ε>0,■ rst choose a real― valued periodic functionん such that‖ ∫1く (ε /2)9and then choose a polynolnial o M″ ith the property

・
降0陣 Cが 引く:h‖ く同υ

3.2. The ring of finite integral adeles

55

The composition Q o h is also a periodic function, and moreover,Tn(Qoh):QoTn@)::Qoi.
Next we use inequalities

(i)
(ii)
(iii)

la-U1u(lac-6e1 ina,b)0,(a + b)o ( 2q-t @q + bq) in a,b ) 0,|max(O, a) - max(0, b)l < lo - bl,

if o, b € IR,

to estimate the following
lf (n) - Qft@))l'<{ltw - max(0. h(nf/ol+lmax(o.

h(n)1'/o - eft@Dl}'( 2q-r {VOf - htu)l + lmaxto. h(nyrto - A(h("))l'}<- 2s-r {lttrtn - h(n\r + (9)'}t"'" \2/ ).

Now taking the Besicovitch q-norm, we havellf-Aohlln{e,which implies

llTnff) - Tn(Q o h)lllc ( e.
The same argument as above, with noting that lt(r)l < Ilhll,, r e 2, yields| /E\s)l?('t - Q(fr@;lc s 2e-r {lOeY
q- -"h\t)l+ \) |Then, integrating the above with respect to ), we see:2a-,{llfn(:'n'",1 f:)'}
lll-aofilln,,{|f,,La(2q-r
lrru -fril,,+\2/ ) - t," -ftllr+ \zJ i<u"'Hence,

llrnff) -0llr, < ll"r(/) - rn(Q o h)llry + llQ " i -9117" 4zr.

The proof is complete by letting e -+ 0.

IFor each n € N, let rr,(r), r €2, denote the smallest positive residue of r modulo n,that is, rn(r): r mod n,if r mod n > 0, and rn(r): n,if r mod n:0.

Proposition 3.16. The functi,on /: N -+ C is q-li,mi,t-periodi,c, i,f and only if for euerg
sequence {r*}u C N conuergi,ng to0 i,nZ, llf- f (r"r(.))lln -+ 0 as k -+ oo.

'

Proof. The sufficient condition is obvious since the functions /(r,,.(.)) are periodic. For
the necessary condition, assume that / is q-limit-periodic. Let {np)p C N be a sequence
converging to 0 in Z. This means that for every fixed rn € N, all except finitely many nk
are multiples of m. Hence it is enough to show that for any given e ) 0, there is a positive
integer rn satisfying

llf - f(rm(.))lln <e for attt>.7.
To see this, first choose a periodic function 9 near ,f , say, llf - glln ( e/3. Let a be aperiod of g. Then by the definition of the Besicovitch q-norm, we have
1N,',VT:o*f V@) - g(n)lq : llf -sll; < (;)'

ft:

l

56

Chapter 3. Limit-periodic Arithmetical Functions

Therefore, there exists an integer lllvhich can be chosen tt a lnultiple of α, such
∈ バT, 、

thatｔｉfollo、vs that

ｂ

一３

く

ε
２

力
ヽ州︱ノ

η

一

ｇ

η

222(・

∫

ノ(γ ι )) gllc=
‖

ｈ〒ん祠
１一
ｈ
＼
／
１
′
︲

Since both f (rL^()) and g are periodic with period

ｈ

・
→
悧 壕が
嘉
09b洲 列羞Ⅸ
all l》 1.

Consequently,

llf (:'m('))- fllq ( ll/("2-(')) -glln+ llg- /llq < e for attt27,

which completes the proof.

n

Let fir be the space of functions f t ? - C for which f (rno@D l- /(r) for somesequence {ru}n c N converging to 0 in ? [39, Proposition 10]. From our viewpoint_, forr > t, the space 5' is just the space L'(2, with restricted condition that "f lx : T;'(f).
^)
More exactly, Iet f e L'(2,)) and 1 t:!;t(f)e D'. Since )(N):0, there is a function/' in the class of equivalent functions of / whose values on N are assrrmed to coincide with/. Then, by Proposition 3.16, f' e $'. Therefore, Proposition 20 and Proposition 26 in[39] can now be rewritten as follows (see also [28,36]).

Proposition3.17. Letf €.Dq and,?:fnj). Thenthefollowingthreestatementshold,.
(i)lNMl/l :: N_+mf(rl : Elll.
.li- *f
1V .Lt "n:l(ii) For (s,l) e R2,

I^/)] : Elexp (J=sRef+ Jltt-D]

Mfexp (J=tsRe/ + Jlt(ノ ,CαRθ mα rん)=(∫ ,亀 ), α∈Q/Z・(3.動

3.18.Proposition 3.17(ii)claims that every limit― periodic arithmetical fllnction∫∈D9has lilnit distribution and its lilnit distribution coincides、 vith the distribution of
ノ=場 (ノ )∈ Z9(Z,入 ).
Theorem 3.19.Zθ ι∫ ∈ Z9(Z,λjηcs.
Ω⊂Z υttん tん θメθ:Jο υ θP"pc焼 う)・

5助 Cη ιんθtt θ

",stsαル ιJ mθ αs包 宅 sん ♯ れ υαれαηιsθ ιθιιχ∈Ω,υ ん留ん(2):=∫ (χ +2),2∈ N・(1)ん ∈D9ル α(ii)τ Кん)=∫ (χ 十・

ι
χ∈Ω
・
)ル αι

(iii)

ん 9=O πηψrmり れΩ
Ωん ‖
∈
ズ
朝増″
〃
.

3.2. The ring of finite integral adeles

57Pη ィ Let f∈ ι9(2,2)and{ヌ m)}れ be a sequence of periOdic fllnctions con℃ rging to f
→ χ +l iS ergodic(Theorem 3.13).Then fOr each m∈ N,byRecall that the shift Z∋ χ 卜
the ergodic theorenl,there exists a full measure shift invariant set

J塩Ωηl such that

リ
″
Ω
十
・
ヌ
η
,χ ∈
o+→ ―
ず=E]β O― Л
ス
判
ゞ
む
井

(3.9)

n=1

Let Ω=∩ m Ωm.It is clear that Ωis shift invariant and λ(Ω )=1.It now follows from
(3.② that

llf:^) - f,llq:ll1^) -fll",, re{r,(3.10)

dmd輔 均 =β りし→ 呻 ∬。

諸群 《 trrおl(3.10),We seeん ∈D9and=ズ ん)=ノ ("十出 鷲 緊 詰 翼 1鱒 よ、

e)for all χ ∈ Ω.

Next we prove(iil).Given an ε>0,there exists an m such that

FOカ タ<:.
Now,due tO the shift invariant property of the Haar llleasure λ
,

FOo+う

リーЛ夕<]"∈ 2
R"十 ・l■ 9=ば0n the other hand,the functionヌ m)is uniformly continuous since 2 is compact.Thus,

δ>O such that lヌ 772)(")一 βm)(ν )￨<ε /3 when ρ(",ν )<δ・It f0110ws thatm)(2)￨<ε
m)‖
m)̲月
しくε
l男
)― 月
/3 for all ηc N,and hence ll厖
/3,if ρ )<δ・
Finally,fOr″∈Ωwith ρ )<δ ,We ha、ん一ん19く ‖‖ん―厖772)9+‖ 月π)一 月m)19+‖ 巧m)̲ん 9
there is a

″
)(η

(・ ,ν

,ν

re

(χ ,ν

ε

□

TheOrem 3.19 is thus proved.
RC772α 帰一Ｑ υ夕 <:+:

十0‑巧 → 隔 十 Fa̲Лε

く ば ― β OL9+陽3.20.Letノ ∈ι9(Z,ハ ).Then by Theorem 3.19(面 ),the limitΩ几0ん =:ノ ∈D9」exists.MOretter,we ha、 re■ (ハ =ノ.

TheOrem 3.21.五 ct f∈ Z9(2,入 )α η
α∫=71(う・乃r θ
αcん δ>0,Pat B(δ ):={″ ∈2:ιρ(0,χ )<δ }.ι θN
鼠→=為 兎⑥ 十→
Xの ,η ∈Rη

Tん cη ,んθSδ → 0.
∈D9α ηαtt Cθ ηυηCS tθ ル ηD9 α

P知 9′ For each δ>0,using Hё lder's inequality,we have

=為 鷹い 勲
臓
州
J
(の

一

δ

λ

︲′

Ｂ

ヽ︑

χ
α

λ

χ

η

＋

ズ
︿

／

く
為 に。

Chapter 3. Limit-periodic Arithmetical Functions

58

It follows that

脆OFく

お

白η+矧 卸 →=お鬼。

=満 ち繭‰ いう
Xの

η+矧 ‰

か

OXの

,

and hence,

mド
がく
中→碗 剛
井
お場
Ё臓
←と
Note that the integrand of the aboК integralis bounded by lノ(χ )9。

MOreover,

Ⅳ
→
鳴
X刷 器
聯初→
井
Ё

り the ergOdic theorem(χ tt χ‑l is ergodic,Theorem 3.13).ThuS,letting̲Ar→ ∞ in
(3.11)and using Lebesgue's dominated con℃ rgence theorem,祀 arri(atノ ・力 9く ‖‖￨IL・

The above inequality guarantees that tt iS a 9̲limit― periodic fllnction.Indeed,let{β り}772

be a sequence ofperiodic fllnctions converging to fin Z9(2,λand耐 → ― 力 ‖9く ‖β→ 一 Л 夕 → 0‑17二)・Then ttm)is also periodic)嗣 bws血 乱

For each m,fron■ the uniforllll continuity

Oo F→ 釧≪
OoKの
―
β
r° け→
房
∠
為
B(δ )

→

o as

δ―〉O unifonnly in η.

πConsequentし ,￨1月 η)― ノ(m)￨19く 1外 )一 ∫い )￨￨し → O aS δ → 0・ In addtbn,鴫 hⅣ e thefollo、ving estilnatem)̲ノ )9+ノ (m)一ノ
カー月272)￨19+‖ 月
角―ノ9く ‖‖772)̲∫m)￨IL9+‖く211f― β湾(π￨19(772)￨19.

Therefore,

limsup lん ―/9く 2‖δ→ 0「―βm)￨IL9.

The proof is complete by letting η2‑■ 〉∞ .

□E"α ttPι θ3.22.Letた ∈{2,3,… .}。 Let∫ :N→ {0,1}be the indicator ftlnction of the set
th po、 ver free integers,that is,ofた ―則=仕 簿:=跳 篤 ニ
踏:メand let∫ :Z― →{0,1}be a natural extension ofノdeined by

肋‐重跡
仕路:=誼 i驚 メ

59

The ring of finite integral adeles

Then f=13,where B=∩ P(2ヽ Pた 2)∈ β(2).It is well known thatノ ∈D9fOr alllく 9<∞ .MOreover,■ (∫ )=ノ ,as vtt will see in Section 3.4.For m∈ N,put■m:=鶴B(δ )tO

Z={鶴 χ :″ ∈ Z}・ For any sequence{η ι}ι COnVerging to O in Z,by changing
Иπ′in the proof of Theorem 3.21,we also ha′ e

九し)=轟 九 +→ λ
ω蜘/as J→ ∞ιAη.

Next, we calculate {f",}t for a special sequence n7 :=Π p<a pι :=Π :=lp:,where pι denotes
the l-th prime. We have

九 0=λ(B∩ (■ ηιtt η))=λ(■ πι
)・

The numerator can now be expressed as

商

ヽ刈ノ

η

η
＋

Ｍ

+→

Pた

The secOnd last equality holds since the sets(∩

are independent. Fix

ι≫た。For each Pく
スη

̀+η

pι ,ヽ

Pく Pι (2ヽpん 2)∩ (■

／

商
＜

入
∩
の
に
ム
陶
ι
範
)鳥

１ ︲ ′

に

十

頃

∩
＞
咳

↓

ａ
︿
メ

Ｐ
︑
ヽ

η
И

ヽ
︵
Ｚ
＜

∩

＜

=入

ヽ

/

=λ

¨笏

\p

∩物
Щ
∩
物
＞

: x (nfZ\ pril) e, (An,* , ))＜^(Bn (A,, *n ))(1‑井 )・ηl+η ))and{(2＼ pた 2)}p>p:

Ve see2+η ⊂pた 2+η=η ι

たIt fol10ws that inι tt η⊂ (Zヽ ρ2),if ηmod Pた.

≠0,while(スnι+2)∩ (2、 Pた 2)=o,otherwise. Therefore,

れ出朝メ
=傷 η
物
吼切
附
￡
̀い

Consequentlェ

←討 '路 譜∫=僣 物
原→附

In partた ular,九 ′(2)=Π >2(1‑力
ρ

),fη

お ん―
th power iee.Thus,{九 !(η )}̀COnVergestOノ (2)unifOrmly on the set ofん ―th power iee integers.However,the convergence is not

unifOrm on N at all.Indeed,gi■ 7en any ηJ,tabΠ P>2(1‑歩1く 9<∞),WhChおη =p揮

not near∫ (2)=0.HOttVer,{九as stated above.

)=
COnК rges toノ h D9 br」 1

1,fOr example.Then ttι
ι}ι

(電Chapter 3. Lilnit― periodic ArithmetiCal Functions

3.3

Convergence of Fourier expansiOns

3.3.l

Convergence of Fourier expansions in ι9(Z,λ )For each η∈N,deine an Operttor Sη :L9(Z,λ )一 〉Z9(Z,入 )as税(め =Σ (λ ち/→ 亀 「 ∈Z9ク ,け
/ゎ

rれ

;

1く α
くτ;(α ,7)=1

1t is easy to see that Sn is a continuous linear operator and has the■nite operator norm

島 ‖ The bll側 hg bmmaお dmihr to p6,Theorem H.1.1].
‖
9・

Lemma 3.23.Zθ ι{η た}ん ⊂ N bθ αSθ 9zθ ηCC Cθ ηυθη れ θ ιθ Oれ Z.Tん θη ιんθル ι υれ θ ιυθ
Jθ

6θ ηα
づ
ι
づ
θηs

υαJθ ηιメ
αrc e9し づ

JJノ ∈五(1)力 r α

9(Z,λ

),

sた → ∞;
乱た‖(ノ )一 ∫
￨IL9→ O α(ii)ι(3.12)

ι
た
ん
α
ι‖
ι
tt Cχ づsts αcθ η αηιK9 sacん ιんθ・乱た￨19く κ9ル r αsι(1)→ (li)・ If the sequence{島 た(ハ }た COnVerges toノ ,then{乱 ん(∫ )}ん iS bOunded forevery∫ ∈Z9(Z,λ )・ Therefore,￨ISれ た9 iS uniformly bounded by the uniform boundednessPη げprinciple.

^ΛLetノ c Z9(Z,入Given an ε>0,there is a periodic fllnction a such that(iり →―σL9く ε}た/(κ 9+1).Let 772 be a period of a・ Recall that the convergence of{η た
(i)・

)・

ll∫

ilnplies that all except initely llllany ηんare ll■ ultiples Of 7孔 。No、v,if ηんis a lnultiple of 777,then Sれ た(D=σ ,and hence

島た(ノ )一 島た
(0+す ―ノILq‖
IL9=‖ 島ん(∫ )一 ノく￨1島 た(∫ )一 島た
(DIIL9+σ ―∫￨IL9

幾希 十
希 毛

v/hich completes the prOof.

□

Lenllna 3.24.Fθ r aι ιη ∈ N,

乱 ￨11く 6,
‖
島 2く
‖
1・

υ
Crν
じ
α
づ
ιηげ ZCmmα 3.23(ii)ル r θοtts,Xl=6 αηdスち=l Sα ι
ι
ん
cr υ
れθ
功 tん c Cθ η
η
υ
Z.
Cθ
CTjη
ι
θ
Oれ
sc9acnθ θ{η た
θ
}ん
jο

2(2,λ
Pη げ Since{亀 }α ∈),it h。 lds thatQ/z iS an orthonormal basis of ι

鳥 (DIIL2く ￨IЛ IL2, f∈ E2(2,λ ), η∈N.
‖

Therefore,￨IS,22く l fOr alり ∈N.
Let us■ ow show that‖ S,tllく 6 for all η∈ N.F破 an η∈N and let f∈ Zl(2,λ ).It
follows from the deinition of Sn that
(ノ ,砲 /η)=(島 (/),a/2), 1く αくη,613.3. Convergence of Fourier expansions
and hence, for any periodic function g with period n, we have\f,o): (^9"1.0,a1
Assume first that the function / is real. fnen .9"(/) is real, too. Let A : {r e 2 :3"(f)@) )0). Itisclearthatla(r) isaperiodicfunctionwithperiodn. Hence,wehave
the following estimate
f t^/lnll,9, t Dt,) - l(')l'l .r(a'):l/,rn{3'tfr-i>o}\"'"" -.1(r)).rtart'/^ (s"t"it(rt
I

/^

t

^

/^

,;.,
.\,..
)(dr)
^
* lonls"tft-f<oj (/('l - s"(/)(r)): J,qI G,ritr*l+2 |-.tr'l))(dr)'/^ (fet-.9"(.i)(')).rra")
t"
/
\ ""
Jen13,(f)-f.01<t,9"f/r -f,to)+2 1^ f1"Sx1a"1(/)-/<0},4 n{sn

I t^

|

/ t^

<0+2 / l/(')lI \@r):2 JAt| l/(')1.\(dr).
|

JAt

I

By the same argument, we also have

f t^

r

f r^

|

<2 I l/(r)l ),(dr).
/ ls,(/)(r) - I@)1.\(dr)I
Jrt
Jacl""lCombining the above inequalities) we see

113,ril - f|r,

: Jnt
* Jrr
I ls,Gtr,t - f@tlxton[ l3"t.Or'r - ftill.r1a'y
II/ t t^ r
f r^ r\+<2{/l/(')lXdr)
I
lft*ll)idrl
l:2llfllL'.
I
|
JArt
\JAr
/

Consequently,

tt^9"r.Dllr' < 11,9"( h- fll",+ llillr' < 3llfllr'.

In the general case, write the complex-valuecl function /as f :0 + t/-i, where j andi
are real-valued functions. Then S"(f) : S"(0)+ r/-I S"(h), and hence the above estimate
implieslls,(/)llz,'(lls"(f)llz,'+lls"(h)llz,'<s(llAlll,'+llhlll,') <6ll/llz,'.

'

This means that ll^9"111 ( 6. The proof is complete.

nNext, we will use the theory of interpolation of norms and of linear operators to show
the existence of a const ant Kn satisfying the condition of Lemma 3.23(ii) for any 1 < q < 2.
We will apply the following two results.

Lemma 3.25 (126, Theorem IV.1.2]). Let B (resp. B') be a normed linear space withtwo cons'istent norms ll'lls andll'll1 (resp. ll.ll'o anall'll). L"tll.ll" (resp. ll'llt.) denote ther,nterpolating norrn,O < a { L Let S be a, li,near transformati,on from B to Bt which i,sbounded as(s, ll.lli)g (8,, ll.lli), r:0,1.

Chapter 3. Limit-periodic Arithmetical Functions

62

Then S 'is bounded as(8, ll.ll") -!r (s', ll.ll;),and i,ts norm llSll" sati,sfies

llsll" < llsllJ-"llsll?
Lemma 3.26 ([26, Theorem IV.1.3]). Let (X,u) be a nrcasure space, B: LLIL*(r),and let 1 { po ( pr ( oo. Let ll.ll1 denote the norms i,nduced on B by Lpi(u), an,d let
ll.ll" denote the i,nterpolati,ng norms. Then ll.ll" coi,ncides wr,th the norm i,nduced on B byLP'(u\ where
να =

p021

poa+pt(l-o-)(=ム ,ゲ a=∞ ).

Let(■ ,ν )=(Z,λ )and B=L∞ (Z,λ )in Lemma 3.26,moreover let pO=l and Pl=2.
9 norm‖・Then for l<9<2,the interpolating norm‖ ・I L9,Where￨lα COincides with the五α=2(1‑1/9).
Let B=B′ =Z°°(Z,入 )in Lemma 3.25 and‖ 。￨lo=‖ ・16=‖ ・ Ll; ・ 1=‖・ 1=‖ 。￨IL2,

respectively Then the norms‖ ・￨lo and‖ ・￨1l are COnsistent[26,IV.3.11.F破 an η ∈ N,
and let S=Sπ 3.Lemma 3.24 claims that‖ SI10く 6 and llS l く 1.Now,applying
Lemma 3.25 to α=2(1‑1/9),We See

Sα く61α ・lα =62/91.
‖
Since the interpolating norms ll。

￨lα

=‖・
￨IL9,it f01lows that￨la COincide with the Z9 norm‖ ・Sl α‖
島2(D IL9=‖ S(Dlα く ‖‖Л L9く 62/91/ IL9,「 ∈Z∞ (2,λ ).

This implies‖ Sη l19く 62/9‑l because L∞ (2,λ )is dense in Z9(2,λ)・Therefore,for l<9<2,constants K9=62/9 1 satistt the cOndition of Lemma 3.23(ii).

Ne難 ,conSider P,9>l Suclthat 1/p+1/9=1・

It is、 vell known that the dual space

Of Zp(Z,入 )COincides with Z9(Z,λ ).MOreover,it is easy to veritt that the dual map ofth詣 ),where詣 )and詣 )denote the same operttor a but on dttrent

spaces LP(Z,入 )and Z9(Z,入")cdnddes宙)・It now f01lows fl・ om the duality that島￨♭ =‖ 島‖‖
9・

TherefOre,constants K9 in Lemma 3.23(ii)e対 St fOr all lく

9<∞ .COnsequently for any

sequence{η た}た ⊂ N COnverging to O in Z and for any∫ ∈ Z9(Z,入 )(1く 9<OO),n‐ e havesた → ∞
乱た(∫ )一 ∫‖
￨IL9→ O α.

(3.13)

3..3.2 Convergence of Fourier expansions of limit-periodic functions
Now we turn to the study of Fourier expansions of limit-periodic functions. For eachn € N, define Sr; Dq -+ Dq as
Sπ (ノ ):=(∫ ,Cα /r)cαrπ/r, /∈ D9・;

1く αくγ;(α ,r)=1

Letノ ∈DC and∫ :=7b(ノ )∈ Zg(Z,λ ).Recall that by Proposition 3.17,we hⅣ e(∫ ,Cα/r)=(ノ ,亀 /r), T,α ∈N.

3.3. Convergence of Fourier expansions
Consequently, Tn(S"(f)) : S"(/). Therefore, for any n € N,lls"(/) - flln: lls"t.fll -fll"".
The above equality, together with (3.13), gives us the convergence of Fourier expansionsof limit-periodic functions, namelyl

Theorem 3.27. Let {nn}* C N be a sequence conuerg'ing to 0 i,nZ. Then for euery

functi,on f i,n Dq (1 ( q ( *), we haue(/) - /lln : o'&lj1 lls"-(3.14)

3.3.3 Convergence of Fourier expansions of almost-even functions
Let cr,r : 7,2, . . . , be the Ramanujan sums,

e,(n)::

eo/,(n).

t1(a(r,(a,r):1

Let E denote the space of even functions. Then, by [43, Theorem IV.1.1],

E : Linc fc, : r : 7,2,...1.
Denote by $a the linear closure of E with respect to ll.lls. Functions in Eq are called
q-almost-even arithmetical functions. Moreover, {p(r)-t/2rr}r:r,r,... is an orthonormal
basis of the "Hilbert space" !32, where rp(r) is the Euler function, p(r) : #{1 < o ( r :(o,r) :1). For almost-even functions, the following result is a version of the convergenceof Fourier expansions.

Corollary 3,28. Let {n*}* C N 6e a sequence conuerg'ing to 0 in 2. Then for euery
function f mtsq (1 ( q ( *), we haue

H- ll tk+crcll z-J ",(f) ",rlnk

where a,(f) :function f{p(")}-t(f,"r),r :7,2,...,- "fll : o,

(3.1b)

llq

denote the Ramanujan coeffici,ents of the.

Proof. Since Eq C Dq, the proof is complete, if we can show that for any .f € Eq ,\-L",U) c, : Sno(f).

(3.16)

rlnk

Let us first show that (3.16) holds for f e ,B.Take / € E of the form

f :T b,,,.

ｓ
ｉ
ｗ
≫¨﹈
ｒ
ｅ
軌
ｈ
ｔ
ｏ

r€I

・
ｒ
Ｃ

一
一

ｅ

劉

ｅ

ｈ

Ｗ

一一

ヽノ

α

ｌ

く

・

１

ω

ｈ

ｅ

ｂ
Ｏ

６

３

０

Ｓ

Ｈ
Ｒ

ｅ
ｈ

Ｌ

ｒ

り
ヽ

・
ｅ

ｔ

舛 ０︐

％

螂満た

Ｓ

︒
ｔ
Ｈ
・ Ｍ 仏
﹂

where 1 c N is a finite set and {br}rer C C. Then a,(f) : br, if r € -I, and ar(f) : 0,otherwise. Thus.

Chapter 3. Limit-periodic Arithmetical Functions

64

Ｈ
．
Σ¨Ｌ

％

し

一
一

К

６

３

Ｓ

ｆ
ｏ

争
い

Σ 珈Σ

一
一

θ

■υ

く

Σ¨

R.H.S.of(3.16)=

Σ珈

Therefore,I)rlnpThis means that (3.16) holds for any "F e E. Since both sides of (3.16) are continuous
operators with respectto f , it follows that (3.16) holds for any f e |B'1.
!
Erample 3.29. Consider again the indicator function / of the set of k-th power free integers.
Define /': N -+ lR as.f'(r) :I ,trlf 1;;.

aln

where p denotes the Mobius function. Recall that / is multiplicative: /(p1) : 1, if I < k,and f (pI):0, otherwise. Then // is also multiplicative. Moreover, for each prime p and

l>7,

(

't

I

u,

:tt-t-

f'(p'): p(t)f (pt)+ t'@)f (pt-t): f (p') - f (pl-t): {^-'' " l- ^lotnerwrse.

Thus,

oo lrl//*\l/lfl/-\llttt^2\ta:?++-[I ( 1a u_]t.!J
I!+ n : LL
p
pz
\
vv

\

/

/

\

.
) : -:l-T { I+ +p',/ )
1\-(3.r7)=,
Therefore, by [43, Theorem VIII.2.1], the Ramanujan coefficients o"(/) are equal toa,(f): t
ol5X.'ff'"'+

Ler {n} C N be a sequence converging to 0 inL. Since / € ,8q(1 ( q ( oo) (seeLemma 3.33), for each n1, the Fourier expansion coincides with the Ramanujan expansion
as we have shown in the proof of Corollary 3.28,

Sn,(.f)(n):to,(f)c,(n):fc,(n)
rtnt

石
ヽ
ヽ
e haverlnt

t ry:
t ry f c,tu).
*l(d<oo *rlnl:r ct,!11".ff,,

ΣECr(2)={:1 11lrllise.

(3.1助No、v,the 1/1ё bius inversion formula implies thatズ→=澪 /° =1ユ 禦澪。
0・ThereforeA→

乳00=1ュ 禦(平 %O Jttμ %°

)・

09

65

J.4. Additive and multiplicative arithmetical functions
Recall that we have the convergence of Ramanujan expansions as follows

as l-+oo.

llS",(/) -/lln+0

On the other hand, we can show that {S",(/)(n)h converges pointwise to f (n) as I -+ oo.
Indeed, fix n € N. It follows easily from (3.18) that
Cr師 )く

澪

η
brttα

.

Let D(nt): mina{d I gcd(n1,d)}. Then D(") -+ oo as / -+ oo. Moreover, we haveげ0‑乳 001=D属 禦(平 %0‑源 °
<∞

)

ldの

→∞
く Σ 鳴型 →O as ι
2η

.

D(η ι
)く d<∞

Note that the condition(3.17)is enOugh to ensure the pointwise convergence of Sれ ι(∫ )(η ).

Comparing with i43,Theorem VHI.2.1(市 )],We need an additional condition to ensureαrげ )Crぃ )=ル );η ∈
Σ
r=1

N・

3.4 Lilllllit… periodic additive and multiplicative arithmetical

functions
A fllnctiOn/:N→ C is called additive(resp・ multiplicatiК )ifノ(mη )=∫ (m)十 ノ(η )(resp・ ノ(mη )=∫ (m)ノ (η ))Whenever(m,η )=1・Let∫ be an additi((resp.a multiplicative)functiOn.For each prime p,deine a fllnction

ん aS

ん(η )=∫ (pん ), if η∈Pた Nヽ Pた+lN, た=0,1,… …

Then the functionノ can be expressed byル)=Σ ん。)。

6・ル)=Π ルし

esp・

))・

2の

NOte that for each η ∈ N,the ininite sum(resp.product)iS actually a nnite one.NOw,
extendル tO a random variableん :Z→ C in a natural wtt as

ら )=∫
(χ

(pん

), if

χ∈ 2ヽ
Pた

Pた

+12,

た=o,1,… …

Then{ル }p iS a Sequence of independent random、 ariables.It is ob宙 ous that if we re―place{ん (η )}p in the expression(3.20)by{ル (・ )}P,then the sum(reSp.product),in
product)cOnttrge?For additive func―

general,is nOt inite at all.Does this sum(resp・
tion,NovOselov[391 proVed the following results.

TheOrel■ 3.30(cf.[39,Proposition 46,471)。

Zθ ιノ ∈

ιんes鶴 %n(″ )R・ ):=Σ ら

p

COη

υ
θ
ttθ υ
cら ■(ノ )=∫・
a.e.■ ∈Z.ν οηcsル rλ ―99 ιC αη αdα をυcル nctづ θη・ Tん cηjιChapter 3. Limit-periodic Arithmetical Functions

The tool used to deal with additive functions is Kolmogov's three series theorem.
For multiplicative functions, Novoselov [39] only gave some sufflcient conditions for which
TIoi, converges. This section deals with multiplicative case and our result is the following.

Theorem 3.3L. Let f e nn be a multi,plicatiue functzon wtthMlfl10. Then the productf@) ,:
conuerges for )-a.e.

lIp frr,lr e 2. Moreouer, fo(f) : f.

The following is a key lemma to show Theorem 3.31.

Lemma 3.32 (cf. [5, Ttreorem 1]). (i) Let {X"}" be a sequence of i,ndependent compler
rand,om uaTiables wi,thBlx,) f o. Assume that ll" E[lx"l] conuerges andffnE[x"]conuerges to a non-zero li,mi,t. Then X :: ff' Xn conuerges almost surely. Moreouer,

X i,s i,ntegrable andEilxll < II"il""ll.
TL(1i) If, i,n addi,ti,on, llnnllx"lql conuerges for q ) 7, thenE[x] :

ll"t""l, Ellxlnl : fltllx"lnl.

P"げ ←)Let
鴫 ‐

週 努 諭

・

Then{■ 42}れ iS a martingale.It follows frOm our assumptions thatE‖

銑I=暑詈宰計,η ∈N,

is bounded.Therefore,by Doob's marting』 e convergence theorem(TheOrem O.40),thelimit

if:=lim Afれ
e対 sts almost surely.Consequently,the limit X:=Πχηalso e対 sts almost surely Next,
π」

applying Fatou'slemma,we see

E[χ l]く lれ fEIΠ χ たII=浮
麓

た=1

塩

Π

EIχた]=Π EIXπ ].

た=lη(il)If,in addition,Π η EllXnlql COnVerges fo■ 9>1,then{EIIA421q]}η iS bOunded.It
follows fI・ Om

Theorem O.40(面 )that(■イη}η COnverges to"「 in Z9。 Consequently,
E1/1f]=駆戦Elllf72]=1,

到 M問 =鳳 到 鴫 問 =M・
The proofis complete by noting that X=̲lf].
ΠれELχ η□

J.4. Additive and multiplicative arithmetical functions
For each prime P,比 おeasy to see that the random耽 Hableん iS integrable,f and only

井)=← ―
じ―DttΨ <唸
かり
一 ・
=かめ ―
可
廟
耳
島
戸
)=← :)ニ ザ

The expectation can be expressed by(多

The folloヽ Ving properties of multiplicative functions、vith non― zero llllean Value can be fOund

in[43,Chapter VIIl.

Lemma 3.33.Zθ ιノ ∈ 99 ιθαmaJォ ゎ εα

ttjυ

Jづ

Jο υづ
ηθ
θル ηθιグθη υjサ んMIノ ]≠ 0・ 動 θη ttθ ル ιんθι
αs.
(i)∫ C ttg.
(ii)

M団 =平

一
←

;)(1+響

+ザ

ー

)=平

到力・

L pα ttCttι αr,ル r θυθη pr,mc,E[ぁ I≠ 0.
7・

P宅 げげ T71θ θ
宅鶴 3.31.Since 99⊂ 01,it is enough to proК Theorem 3.31 for the case
9=1.Let∫ ∈01 be a multiplicative function with VII[メ I≠ 0・ Then l∫ ￨∈ つl WithNl[げ ]≫ INII[ノ ]￨>0・ MOreover,the ftlnction lノ l iS alSO multiplicatiК andノ IP=￨ん
Now,applying Lemma 3.33(ii)tO fllnctiOns∫ and lノ ￨,We have the following。
￨・(i)The product ΠP E[￨ル I cOnverges.
lii)The productΠPElル ]conК rges to the nOn― zero vdue M『1・These mean that the sequence{ル }P SatiSies the condition of Lemma 3.32(i).Therefore,
2.
a.e.χ ∈
A→ :=Π ら
し),入 ―
P.ThiS pr00f is divided into three steps.
Next,we will proヽ 配that Tl(ノ )預 ノStの 1.Чhen g>1,MIIノ 1判 =E[￨ノ q and the Ramanttan coettdents of/cdncide wtth
those of /, that is,(3.21)(ノ ,θ r)=(ノ ,3), r∈ N,

where a:=7L(cr).Indeed,ノ ∈99 implies∫ 19∈ 役 Then Lemma 3.33(li)applying for
1・l∫ΠpEIIル

19 implies the cOnvergence of the product19]・

Therefore,by Lemma 3.32(li),=Π EIIら 判=MIノ 判

E‖ Л判

.

(3.22)p

Next,we will prove(3.21).It f0110ws from the f01lowing expression of the Ramantlian sum
Cr(η

)=Σ

ttπ

(γ /ご )α(γ /ご )ご =Σ dγ μdr μl、 (2),that

ld20)・らし)=Σ μ/の α。
αlrChapter 3. Limit-periodic Arithmetical Functions

68

Therefore,(3.21)will h01d,if(∫(3.23),ldN)=(ノ ,ld2), α∈N。

Let us prove(3.23).Fix α∈N and write d=Π p∈ r pα (p),where f is the set of primes P
such that α(p)>0.Then(∫ ,ldN)iS represented as([43,Theorem VHI.4.41)

協ヾ =M1/1碁

絆

件

十
磐

+…

早

)(1+響

+撃

一

)‐

,

or it can be rewritten as

げ,1ハ )=Π E[ら
p∈

lpα

f

Oメ

ΠE[ら

].

p/1

The sequence{ん }Pグ r alsO Satisies the condition of Lemma 3.32(ii)beCause r is a inite
set.Therefore,

EIΠ ら]=Π EIら ].
p￠

I

P/1

This implies that(スld2)=E[Π んΠlPα o月 =E[Π らlpα o2Π λ
p

p∈ f

p∈ I

P

]

I

12α o月 E[Π ん=E[Π ら]=Π EIら lpα Oメ ΠE[ん ]=げ ,ld●
p∈

r

p￠ r

p∈ I

.

P/1

石
Stの 2.ヽ ヽe will pro(■ (∫ )=ノ in the special ctte 9=2.Since the ftlnction∫a multiplicatiК fllnction with non― zero mean value,Lemma 3.33(i)implies thatノ∈92 is∈磐2.

Moreover,{9(r)‑1/2cr)r∈ N iS an orthonOrmal basis of the Hilbert space磐 2,thuS we ha/eParseval's identity tt follows:Ψ 珊 =Ⅶ 麻It then follo、 vs that

魯鼎 柵 f蝸

Note thatャ (r)1/2a}r∈ Nぉ an orthonormalsystem in L2(2,入 ).Now Parse稔 1's identitbs
fOrノ andノ imply

こ 需Cr些 ノ as E→ ∞

,

R…
a堕「器
需
こ
In addition,it is clear that

乳([11先辞

Cr)=』

11気 禁

a,

J.4. Additive and multiplicative arithmetical functions
and hence, letting R -+ oo, we obtain Tzff) : f.
Step 3. Now we turn back to the case q : 1. Following the first proof of [43, Theorem
VII.4.1], we define the multipticative function fV) Ay truncation of / for K > 2

ρ的=僣ハ 出II
IじThen the following two statements hold.

0 fK e D2.
(ii) ll/(r() - /llr -+ 0 as K -+ x.
The statement(ii) implies that M[/(r()) + O when K is large enough because Mlfl + 0.
Therefore, when 1( is large enough, the natural extension V'o) n"rt9 is well definedand moreover by Step 2, Tr$6)7 : Vn). We also have:(ii)' llflr() - fll", -+ 0 as 1{ -+ oo.
Let us prove^statemen(ii)'. It follows from the defiqition ot fxl that lflK)(")l < l,i(r)1,
and hence lf\K)@)< zlfr")l for )-a.e. r e 2. P*- flllス ):={χ ∈2:ヌ K)(χ )≠ R")}.
(κ

It is clear thatた +12)(λ ―∪ (ρ 2ヽ

4(K)⊂

P考

a・ e。

).

p;た >1;∫(pた )>K

Consequently,

xA(r);gf.*('-;)
G-#)o,D*,,
o,*E,, ^P(pk)l>Klf (pk)l>K1\
I /<;(r-:)
p/ f
l(\

o/r,

trr--krtU)!--!r--as as K-+oo.
p*lf (pk)l>2

where, we have used the fact thatI

Z-/
P; k>-l;lf (pk)l>z'

f,91 . oo, ([43, Theorem VII.5'1, Definition VII.1.2]).
DK

Therefore,lll') - fll",: I4(x) lVn\{,) - ft )l)(dr)J{ Jo,n,2lf (')l'\(dr) + o as 1{ -+ oo'
Thus, the statement(ii)/ has been proved. Finally, the statement(ii) and the statement(ii)/,
together with the continuity of Ty, imply that
: /. The proof of Theorem 3.31 iscomplete.
"r(/)□

Chapter 4

The Distribution ofた ―th PowerFYee Integers
4.I Introduction
For,ke {2,3,...}, let y(x)(Q,n€Z,betheindicatorfunctionof thesetof k-thpower
free integers, that is,7,-(r)(n),:{ l, [];;fn:,,iin,i.ri,

I t-\

and let Sl?'(-), m e Z, denote the frequency of k-th power free integers between m -t 7
and m a ly', that is,A,.slf)r-;,:"N \,,,, ._ +Wi

4,,

x(k\1m\,,( + n1.

Then it is well known that for each m € Z.

,. ^(k), : I

J15 sil'(^)

C&),

(4.1)

where ( is the Riemann zeta function.

Many researchers have been interested in estimating the error Slf ) (-) - 1l e &). Under
the Riemann hypothesis, there is a conjecture about this;

ve >

'

: (xrlz*+')/ . ty' -+ oo.
0. lr (s$'(m)+)<(k)/ o \
\ '"

14.2)

As is mentioned in [40], this conjecture should hold, but it is quite unlikely that it will
be proved in near future, because it is related to the Riemann hypothesis so closely. In
particular, in the case of k : 2, there have been many challenges to this conjecture,assuming the Riemann hypothesis, such as [1, 2, 20, 3B]. Refer to [40] for an overview ofthis topic.
In this chapter, we study the probabilistic aspects of this problem. We take here acompactification method which has been developed in Chapter 3. Let us give an overview
of this chapter.
Recall that2 denotes the ring of finite integral adeles and we consider the probability
space (2,8(2),^). Since X(ft)(n) is a multiplicative limit-periodic function, it has theChapter 4. The Distribution of k-th Power Free Integers

natural extension 1(t)(r) as a random variable on (2,8(2),^). As a consequence ofergodic property of the shift r + r + | on2, we get the following law of large numbers,JTLsjf'(r)::;55#i",*, (r+n): B;7s(*)1 :\-a.e. r#,e2, (4.3)

which is the adelic version of (4.1).
The main aim of this chapter is to study the convergence rate of the law of large
numbers (4.3). With the help of the explicit formula for the random variable S$) gi,rert
in Section 4.2, we can estimate the rate of convergence in Section 4.3 as follows;.
r [ (" ( s,l' -+)q\k)/))'] - const N,/k

L\ \',".

)

Finally, in Section 4.4, the last estimate is translated into the language of Z as

t !

/

/

I \\2
(t(s$'t-)-#))-const'Nt/k (corollarv 4.r7)'
JtXn I
trr, -.;;;J,5,:)." *"' ,"t,'::,:":or the conjecture (42) rtshourd be notedthat we do not need the Riemann hypothesis to prove this and nevertheless get the same
exponent as in the conjecture.

4.2 Explicit formula for S,r,'In what follows, we fix an integer k > 2. It is known,nu, 76(r)in) is multiplicative andytD(Q € Dq for all 1 ( q < oo. However, for the sake of completeness, let us give theproof here. It follows from the definition of X(k)(n) that X(k)(n) is multiplicative andy(t")@):For each .L e N, let Xf)
Indeed, observe that- pex(n)).
fl(rp,: flo<o, Q - prr"(n)). Then Xf) converges 1o;(ft) in Dq.
1x(*)1n;- xy'(n)l ( L, oor(r),PlPrwhich implies.

rIN..rlNI|;rtk)inl-xf)@tl . out'''

p1,(n):
p,r,(n) < I p
I
^I;t
p)pr"
n:lplpr
n:I

#fn:l

p>pL

Thus,

i;xtk)1n; - xf)@)l
} n:I

n:I

a

o as .L _+ oo.
,+ror
1

Since the functions Xf) ar" periodic and converge to X(ft) in the ll.lln norm, it follows
that X(ft) € Dq for all 1 { q < m. Hence as we proved in Section 3.4,

y(n)@):- por"(r)), r e2,
ll(t
p

73

4.2. Explicit formula for S,^v
is the natural extension of X(k)(rr).
If we put

B(0:=∩ (2、 Pた 2)∈ β(2),ρthen it iS Clear that」 κ(た )=13(た ),and thus,叩Ч=入 o均 =平 (1‑井 )=島・Recall that the shift Z∋ χ l→ χ +l iS ergodic(Theorem 3.13).Then applying theergodic theorelll toス「(た ),v″e obtain

SPO=井

羞

XOけ → 彎到XЧ =おい aπ め,which is the adelic ttrsion of(4.1).

lVe are now h a podtbn to JVe the exttctt brlnllla br SPo).島 r eaCh Z∈ N,bt

χlり o)=Π (1‑ヶ(″ )),

pく pL

メし+0,
Σχ
Ⅳ

SttL。 ):=井

れ =1

・
p2ι ∈
ML:={鶴 =pFl・ ・
N:0く αl,...,α Lく L}・
Rcmα rた 4.1. From now on, if there is no confusion,、ve wi1l oIInit(・ )in fOrmulae. For

vill be considered asス「(ん )and s。 。n.
example,」 て 、
Len■ Ina 4。 2.Fθ r cacん Ar∈ N,
SN,L(χ )ヱ 奎 T Slv(χ )● οれ ι211tSc cθ ηυθη Cη CCノ ,=濫 バ0停Sェ ズ→

井(空 鋼 半

(4.4)

二―
鰐μ

)),的

υん
θ
tt μα
んε几
イ
nCtづ θ
ηcnο tθ s ι
σ zsルι
bづ.

P印 げ The cOnttrgence(4.4)is obViOus.ヽ ヽ石e now prtte(4.5).The deinition of SN,L(χ )ヽη︑ ︲ ′ ／

″

＋

たη

ら

２
＋

η

¨
．
＞

ヽ︱ヽ
月ノ

χ

っ

ばけ

た﹁

十

″

ら

一
一

ｌ 一
Ⅳ

た﹁

一
一

ｌ 一
Ⅳ

ら

一
一

１ 一
Ⅳ

Ｌ

χ

鈴

ｌ 一
Ⅳ

ＮΣ 直ⅣΣ 祠 ＮΣ 祠ＮΣ 祠

glves

Chapter 4. The Distribution of k-th Power Free Integers

74

ヽ
η

＼

︑︲′／

χ

十

し
ρ

Ⅳい
々
祠
︐

／

Ⅳ
１一

Ｚ

ｆ ︲ ヽ

μ

Σ咄

(4.6)

Here we have

1gN )_ P$(r *n): ,^,,I r/v*rmodukr
L----;i-l: F| (Nlrtnoduk (l/+r) modzk\\---iF)----;k: ur1 - lr,t1 ((w+r)moduk - rmodzk\---e- )\--

l4'7)

Therefore, substituting e.7) into (4.6), we obtain (a.5). The lemma is proved.

I

The following is a key lemma in this chapter.

Lemma 4.3 (cf. 144, Lernrna 8]). For u,,u € N and A, z €2, we haue

modu xmodu\/(.*r) modu xmodu\ll/(u+r)\v ' */ ---"*?.t /\
u /)
u
u
L\
'

Er ll

_ (ymod (u.u))A(z mod (u.u)) /., _ (ymod (u.u))V(z mod (u.u))\tr.,41,.,}\'-

7

where the erpectatzon E works on r, and,(u, r) : gcd(2, u),{u,u}: Icm(z, u): the least common multi'ple of u and u.
Proof. We divide the proof into four steps.
Step 1. For c,b,c € N with (b,c) :1 and for r e Z, it holds that
1

fi
b4

(r + sac) mod ob : r rn\d oabab +b 2b,.1 .

(4.8)

This is shown in the following way. Since (b, c) : 1, by a similar argument of 127, Theorem56], we have

{ (r + sac) mod ab : s :0, 1,...,b- 1 }: {(r* sa) mod ab : s :0, 1,.. .,b - 1}.
Thus, it is enough to prove (4.8) only fot c : 1. Moreover, we have

{ (" + sa) mod ab : s :0, 1,.. .,b - 7}

: {(, I a*so) mod ab ; s :0, 1,...,b- 1 },so that we have only to prove (4.8) for r:0.,1,...,awe have (r * so,) mod ob : r I so, consequently,

1. But then, for s:0,1,....b-r $tr*sa)mod ab :_ 73t+ s& :b1
Thus (4.8) is valid.

"bx, b-1b1 ,b ob- %'

1,

75

4.2. Explicit formula for ,9,n'
Step 2. By the fact that for z e Z,
(z I sac) mod ri,b : (z mod ab + sac) mod ob,(z mod ob) mod a: z mod atand by Step 1, it is easy to see that for a,b,ceN with (b,c):1 and r,A €2,

1S / @*r*sac) modcb (r*sac) modab\
*
uL^\
)
"b
t ((y*r) modo xmodc\

=t\

-

o

" )

Therefore, for any periodic function f :2 -+ lR with period tzc, we have

/ -r r) mod ab r mod ob\ .-,'lt fL("-;,)f ",@l( ,,[by the shift invariance of .\]sa_c)moaaa _ (z*sac) modab):1F
Bl((y:L r+ ababb/"r,-+.ro.ylIL\

ftH (fu+r*sac) modab (r*sac) modab\",-,1:Ell5t-Lbfu\
nb
ab

/"".l

+ rl moa o r mod a)
: ln
| (@
- (L ./t.,ll')
a
b-L\

/"
Step 3. Set a :: (r, ,), b :: ula, c :: u lo' and / to be?,, (z*r) modu rmodu
JtL')::

-

,

,

'

Then Step 2 implies that
f/
"L\
"

(y.+ r) moa uu-r mod u) ((z + r) moa u

u

, /\

-

r mod u)l

u ll

1(z-l r)modac- rmodac\l:'[\-l /@-lr)modab- .rmodob\
* )1
*
,b /\
,a

1 /tg+r)moda rmodo\ /(z*r)modac rmodac\l:tnl(, - " )( * - * )l2,c,1
By letting y,b,c/(r) in|and

Step 2 beand(y*r) moda_rmoda,
aa
respectively, we see that the last line above is equal to'r:p fl{/rrs*r) moda' - rmoda) ((z+r)moda - "-ooo)l(4.9)
a /) .
a
a /\
a
b."L\
Step l. Without loss of generality, we assume that g mod a ( z mod a. By Corollary 3.7,

the integrand of (4.9) is continuous, and it is periodic with period a. Therefore, Lemma
3.10 implies that
(4.9) :f1r((y+r)

bcoL"^\

noao_smoda) ((z+s) moda _smoda).

a

a

/ \

a

a

/

(4.10)

Chapter 4. The Distribution of k-th Power Free Integers

76

Moreover. it is clear that
¨

ｒ
ｌ

一 α¨

ノ
ヽ

(y+s) moda smoda

︑

一α

︱

if 0(s< a-ymoda,,*1, if a-gmoda(s(4,―Ψ={革 ヒand that

if Oく S<α ― Z modα(z+s)moda

1, if α一 z mod α く s<α,.

Finally, dividing the sum (4.10) into three parts and using the above expressions, we arriveat(4.10)= 尭 :(。

望
三 整ltt・t芦:1・くs<層 mod α

十 Σ T(響 →α―z mod αくs<α ―υ mOdα+α

翌 Ll‑1) (竺 二重‑1))・ 響̲ty=禽 tく s<α (2」 :些

競(Ψ ttla̲2111 →
十

T

(響

+ (Ψ
発 :(ν

‑1)(響

mOdα )(1‑Z modαT‑1)(‐‐lnOd α― ν lnOd α)‑1)(ν lnOd α)))翌(1‑重 〒1+￨二 )

堅
些
⊆
二
主
聖
堅
塾ギ
翌
寄
挙
争
Lニ

(1‑1竺

[)

The lemma is proved.

□

A small modification of 144, Lemma 9] gives the following.

２
引﹁ハLemma 4.4. For any bounded functr.on 11: N -+ R , i,t holds that

＞

型
,0=2堕
。
Щ
Ψ
還lΨ
募←
―
Щ
い
鱚
響
←
二 嘉
還1響
畳
望
μ
し
翼
弓
)=a鴫
言

∞

＜

・

,0)三

Lernina 4.5。 Лθr cα cん zV∈ N,
)(【

2(2,λ
づ
s cθ η
υ
cη θ
η
ι
jη 五
).

竿

鳩

(4.11)

4.3. Estimate of the ,L2-norm and limit points in Z2
Proof. Fix an ,n/ e N. For finite sets lL and Nd such that L c MI C N, Lemma 4.3 andLemma 4.4 imPlY that

EI(恩

些

生

μoば

宗

生 一

ギ

)

二・
一
―
μ
°
)(1壁場
半
等ギ))21】=缶
Lμ恩 ヽ°μ° EI(埜 墜 察 空 生―型 半 二
ん

×

=鶴

)
た

)lnOd υ ̲″ lllod υ((Ar+″

)]

￡
L← ―
型
μ
μ
°
°
響
≒
務
t等ギ
恩Ⅷ
)

Ⅳ
LI協響
属ヽ
仏

旦 → 0器

N篤
忍

L紫

L/N.

等

The lemma is oroved.

By letting Nd I N in the proof of Lemma 4.5, and then IL I N, it follows that

I lr(r) ((JVfl-'*'\+ r) moa ztu

uk-r mod uk g ?(r,.a/) as n- N.
,z-\*'-"/
uk ))(4.12)

lute convergence),(4.lo

On the other hand.

型
言

li下

(abs。

and ,9ry,7, L3 s,√ by the bounded̀:2=ζcon、 Zergence theorem. Therefore, using these con―vergences in the formula(4.5),we ha√ e an explicit formula for Sr)as in the following

theorem.

Theorem 4.6。 乃r θ
υ
θ
ηθ
ι
θル θ
α
んN∈ N,α s α
9Zα んνれ五2(2,入 ),ι ん

ん
ο s,
:ご

̀町

O(些 場 ■ 三 ―篭 等 )・
球)0=島 一井 μ
言
4.3

・0

Estilllllate of the Z2̲nOrIIIl and lillrlit points in Z2

Ls∝ ■。n42 we pO(d tllat N(SNO―

島)=―

Tし ,め and

いこ
狗
面(場塑―
ギ )鰤 政.

Chapter 4. The Distribution of k-th Power Free Integers

78

Let us calculate the explicit formula for the L2-norm of T(r,lf). BV using Lemma 4.3
and Lemma 4.4, we haveEllr(r ru t2l :,rT_, (,?U r r, I (q1{#4 _' Tr.') )']|)moduk-rmoduk)u(u\u(u\((l/+r)
"''\
uK /
uK
Lr?.r''/ (N + r) mod trk r mod uk\ I\-----;-))

rim Ef t=Lt_+us: urT"",Ifi#('nr moa t''')o) (t- " i:lJ? ''-)

l/ mod (u. u)k\
- ; s(u)rrlu) (lr(u, u)k) (, _ --T;if"'-*-oa'*" ' ./ \),-*-*lt. rlr \'"=:4P(.nr-oa"-) ('-tt#4)

1,I(' fr)

(4 1b)

where in the last line, we have applied Lemma 4.4 to

H(n): Hw(n),: (ro mod nk) (r - t 1?o "-)Lettl^\.r,,).-lufu)lilr_HThen / is multiplicative, and moreover the following conditions hold:①平留≠<∞

0花 Ψ

:

<鳴

)平 旦
い
守」<∞

0シ ノ≠‐;

Indeed,by deinitiOn,=毒 =洗 ;ズめ=叩 =2けズ勁Therefore,<∞平留≠=Σ 雨≒万旦学工=Σ 満 <∞平
which pro、 res(i)and(ii)・,,

The Series in(iii)iS O Since∫ (Pι )=0,(ι=2,3,…

.)。

(iV)iS

obvious dnce/お podtiVe.Since/sajdes(i)← V),the mean ttlue M『 l eXiSts and isnon― zero(see[1司 ).MOreOVer,by[43,Lemma VH.1.6],
NII『]=II(1+ノ (p)‑1̲∠皇1)=II(1‑:)(1+」 手]ち)・

4.3. Estimate of the .L2-norm and limit noints in .L2

79

Proposition 4.7. ,4s N -+ oo,
√九

１

ノ
ｔ

リ

一た

ノ

Ｍ

＞

Ⅳ

ー

ヽ ︱ ノ

Ⅳ一
瀞

一

り ヽ ｔ

ノ

ｒ

ｒ

ｒ ｔ ヽ

ｔ

祠

／

ヽ ｔ

η

り ヽ

Ⅳ一
が

ｒ

∫

∞Σ

°
°

{χ

}(1‑{″ }) 山
″1/た +1+0),

where {r} denotes the fracti,onal part o.f r.

Note that

i t,, )'lnr){ +} (' - {+})pr),2"\ tn*)/

Ettr(' N)t2l"r : ll (r- +)i \Theorem 4.8.

ノ

／

︑ ′

︑ ︲ ′

χ

＼

EJ,

((2‑力 )
(27)11「 (力 )Sin兌

Proposition 4.7.

ξ
(2‑力 ){″

1/た +1ヽｎ

1t ∞ }(1‑{″ }) αχ =

ノ

一

ｉ
ｅ
ｈ

０

ｔ

ｅ

ｆ

Lemma 4.9.

血It remains to calculate the分

E[IT(″ ,Ⅳ )121=

一

1/た

ｌ工
／′
＼
︲ヽ

椰

」
塩Ⅳ

■

Ｌ﹈

Consequently, Proposition 4.7 gives us the main result of this section.

(2π )1 l「 (力)Sin,

Proof. The Fourier series of the function {z}(1 一{″ })iS as follows{″}(1{″ })=:一

争ЁΨ
η =1

=争

一

二

翌
ゞ L堅

)

Terllll、 vise

integration yields that
{∬

:プF°

}(1 {″ })αecause

１一
′

望
生
鰐
卜
帯
を
=二 中 ・
こ
=争

′一
６

1,三

∞
▼
ん祠

(手

二χ
χ °
ttЁ
° α
°
″
=:ら
ti留ド
,1争 ズ
∞
=:争 十
解
輸
二 ズだ
碁
∞の
=:ん 二
森ズ済
=:υ

tt・

(°

=耳

「
色
」
理

(:.ア I力

i:三 :￨≒ Tご

:[争
:肯

α
ν
)((2‑l).

=:Cた 。

Chapter 4. The Distribution of k-th Power Free Integers

80

°塾 笙
<2,υ >0)∫∫ 〕群 α"==函 マ号ヂ嘉+≡ 戸(0<包lVe here note that fl・ om a formula

:プ ￨∞

絆

dν

=ノ

F° lν

∞

=ズ

一
:‑l Sin2 ν
ご
ν
‑1)′

(― ν

Sin2 ναν=̲ν isin2ν lF̲ノ

=ズ

一
カ νν
α
ν
ν

∞ =紹 ・
為Tの￨∞(―)2 sincos

Substituting this into the above,、 ve haveC{"}(1‑{χ

})α

:.li°

″三
角

2「 (:)sin g:ξ

c(2‑力 )

(2‑:)
・(27)1 lΓ (l)Sin税

The prOOf is complete.

□

In order tO prove Proposition 4.7,v″ e need the follo、 ving lemma.

Lernma 4。 10.Lθ ι{α η}η bθ α θθ名ヮ ″ sc9鶴 Cη θC.PZt Sれ :=α l十 ・… +α η.ASSamc tん α;
iθ

ι
んθ
"θ

θ sts α θοηSι αηιc∈ C sacん ι
んαι
"づ

S Ⅳ→∞
・
汁→C α

に1の

ⅣΣ霧
・
給→:α S Ⅳ→∞

lη
・

動θ
2,力 r αηνs C(0,∞ ),S

η =Ⅳ

P印 げ Let s"=Σ ιι,(χ ∈R十 ),be an extension of sη as a function on R十 .clearly″αくhm.→ ∞ s″ ル =c・ First,祀 check the cOnvergence of Σ η許 ・ For Ⅳ ,M∈ N,̲lV<χ
,

=け ⇒
π
}罫 可
ガflち 山絆十=け ⇒井―寿ギ∵嘉+#糸・ノキ

81

4.3. Estimate of the ,L2-norm and limit points in -L2

Σれ許 rお COnvergent.Next,lettil■ gM→ ∞ in the above,and thenTllis tells us thatmultiplying thiS by NS yield that

Sf子
S思
Ⅳ
Ⅳ
暑¥
丼ポ
瀞=い ⊃
諷 S十 ⇒

f子 7″まヽ面百:争―

デギ:≒ 雫

:(s*rl, ffi#-ffi?

From Lebesgue's dominated convergence theorem, the assertion follows immediately. n
Proof of Propositi,on 4.7 . Let us consider(' - {{})
i,,"){{}
3,,,", lro J \^ \r* ) )('-{#}): t rr,){+}"1"*J\ ('-{+}).
Ln'J\ tn")/lnr)/ t rht{+}n<Nl/knfr,*"':.F,*/(,){#} ('-{*}) .'",T,_ #-N".n,r#By Lemma 4.10,23 Ar→ ∞
Ⅳ
π澪

9=NVん
/ん

,

声

‐

llA‑7λ十 ,印1:￨)=N聯 (四 4う

π
耳
/た

and

t #)

:111'\tkN'\/k (,*''*,'o-'n>Ntrk" /\It will follow from Lemma 4.7I and Lemma 4.I2 below that>, #:n>r^vr/k

rv2(#q *,(r)) . (4.1e)trh){4} ('- {*}) : ru'/* (rurtrtl l,* ttffi!)a'+otrr) e20)
,G,0"
I n" J

The proof is complete by combining the estimates (4.18) (4.20).

nLemma 4.LL, As ly' -+ m,

η
F/ん

た
▼
量轟型 十
⇒
∝
{#}(1{半 })=Ⅳ (1/∞ 鵠dχ).

Proof. In the proof, we will us Euler-Maclaurin's formula: for any I eCr(la,b]), we haveΣ体一¨
=Ib岬 "← 4‑→ ){+Ib← 分
α
→
分は
9。

Note that Euler-Maclaurin's formula still holds for functions which are continuous on [4, b]and have continuous derivative except at finitely many points' Let
9(・ ):={:羊}(1‑{::}),

χ≫
1・(4.21)

Chapter 4. The Distribution of k-th Power Free Integers

Thenり ヽconthuou&Now,f(占

ヽ
１ ′

一

ノ

／ ｆ ｌ ＼

Ⅳ一メ

十

ヽ︱ ′ ノ一Ⅳ一
メ一／′ ヽ ＼

ヽ １ ︐ ノ

ヽ １ ′ ノ一

Ⅳ一
メ

／ １ １ ＼

一︑ ′ノ

／ １ １ ＼

ヽ

一

Ⅳ一
メ

／ ｆ ｌ ＼

９and henceι,then<χ く(γ )1/た br SOme podive mteger

)1/た

+

p'(x):r(+-,)-ry+ - 4i : (r{+} - t)') 14"+r'")rk+r' -\r*
rfr+l\-[r*)

It follows that g has continuous derivative except at finitely many points in any boundedinterval. For I € N,-L ( l/, by applying Euler-Maclaurin's formula to the function rp witho: (+)t/r und b: Nrlk, and noting thar 9@) : gQ): 0, we obtain
ヽ

ヽ

︱ ′

り

ノ

ｔ

ノ

ｔ

ヽ ｋ ｒ ノ

Ⅳ一
瀞

ｒ

ｒ

一

ｋ

り ヽ

／′１＼

ヽ

ｒ

Ⅳ一
瀞

＜

Ⅳ

Σ％

Ｎ一
Ｌ

=Ib{]￨}(1‑{f:})dχ +ブ[Chan tge variable lνα:)(2{::}‑1)井 "
ん
=弓
=耶 χ
=Arノ
=α → ν.=Z;χ =ιけ
ν
({χ

￨♭

/χ

}―

;χ

‑lMν
十
=半 /L→
ν/L({(1;)1/ん }― :)(2(ν=ギ /L土筆Pの +α り

洋の
r1/ん

1

(1‑― {ν })ご}―

form thatfor五 ,Ar.Chool 3e Z=Z( Ⅳ)SatiSサ ing
Note thatthe山 o( estillnate O(L)iS uni:seeZ(Ⅳ )く ̲lV ;Z(Ⅳ )→ ○。;z(Ⅳ )= 。(Ⅳ 1/た ),Wl
ヽ

一瀞

ｋ ｒ

ゥ
ヽ

Ⅳ

一

り ヽ

ｒ

ｒ

ｋ

ヽ

一ぽ

１ １

ヽ

り

／

Ⅳ

ｒ

Σ

リ

ｋ

ノ

＼

ｔ

ゥ
ヽ

ヽ
ｋ
ｒ

ヽ

一

り

Ⅳ一
瀞

ｒ

ｒ

ｉ

ヽ

ｔ

ヽ

り

／ｒ

ｒ

リ

ｔ

′

＼

ｔ

く

})十

Ｎ

{#}(1‑{幾
た
(轟 ソ
≪
黒μ
則

Ⅳ一
瀞

Σ％

=れ

=0(端 鵠 半ズ 型
α
Ц
り
雅Pの 十

)十
Ц
め
主
=メ ル
力+ο (轟万
評
場
)+0(絆
(:ズ

The pr00fis completeけ letting Ⅳ→ ∞□.

Lemma 4。 12.ス sⅣ → ∞

,

ヽ

ヽ

ｔ

︑ ′
ノ
ｔ ｒ リ

り ヽ

＼

レ
ロ︺ 物

Ｍ ρ
一 ヽ
↓ Σ
ズ
↓
％Щ
ｒｉくｌｋ

(χ

Ｎ一
赫

１ １

一

ｒ

ｔ ｒ リ

ｔ

／

り ヽ

ヽ

ｒ

Ⅳ一
瀞

∫

一

Ｍ

η

ノ

Σ

Proof. Put))・∈ R+)・: o(Nr/k).

Estimate of the ,L2-norm and limit points in /,283Then it is not so difficult to check this formula; for g € Cl([o,b]),1 < a <b < oo, we have

90‑象 の■のα二η)― /bS090山 十
9し

90・

bα

ヽｌノ
≫

﹄

︲
ａ

句

ε

く

ａ
ｈ

島
﹁

ｂ

触
井α

一

リ

a<n(b

Given an 6' > 0, there exists an No such

ヽ ＞ り

停

一

ｒ

ｒ ｉヽ ヽ

ｋ

２

／

ヽ

ｔ

＼

リ

ｋ

ヽ１ ′ ／

り ヽ

ｒ

一

ｔ

１ １

ヽ

η
α

り

Ⅳ一
伊

ｒ

／

ヽ

ｒ

Ⅳ一
瀞

t

↓
Ｓ
ｂ
ｆ ノα

Of course, the above formula still holds if p is continuous and has continuous derivative
except at finitely many points in [a,b]. Now, for,L e N,, < lr, let o: (f)t/k ur.ab : Nr/k. Then by applying the above formula to the function cp defined in (4.21) andnoting that /(4,) : f (b): 0, we get蝸 .If α >蝸 ,祀

have
＼
︑ ′

リ

ノ

ｋ

＼

リ

ｔ

ｒ

ヽ ｔ

Ⅳ一
瀞

ヽ

一

り

１

ｒ

／ １

ヽ

ｒ

り

ｋ

ヽ

ｒ

Ⅳ一
が

η
α

Σ篠

=げ

α

象
→
(2{併 }‑1)井 引

′
ゴ 与
ε
く
科
=響 。 ドりく
響計
刊ル=響 (弁 )°
聯占 ル性
=♂ Ⅳ
d"

Vん

(because lS″ ￨く

i4b」

′
χ
ε
)

A・ l―

1

In addition,η

23α {i斜 }(1‑{ギく Cν =C(1:)1/た:}),

where c=sup αη<cЮ .Now,given ε>0,choose an Z such that c/Z1/た <ε /2.Next,′
1)/たchoose εsatお いng♂
<ε /2.Then there e対 sts an tt aS abOve.For Ⅳ bdng
吾[Z(たα=(γ )1/た >‐lVO,we hⅣ ehrge enough such that

{i争

ヽ

ヽ

ｋ
ｒ

︑ ′

り ヽ

Ⅳ一
が

ｒ

′ ︲

一

／

π

ヽ

リ

ノ

ｔ

＼

ｔ

ヽ そ ｒ リ

ｒ り

α

Ⅳ一
瀞

瞑

十

一

ル

．
一

ル

Ｗ

︑．︲︲′ノ／ヽThe lemma is proved.

ｖ

く:Ar1/た

．＼
／１

η

ｒく ｔ

く

く

い
一
川
叩 ん柱

J/ん

開卸

ｂ ギ

lη

α
η }(1‑{#})+:Ar1/た =ε Ar1/た 。
□

Chapter 4. The Distribution of k-th Power Free Integers

84

玲
0‑議 賓
州=洗 とい ―
島

Let

+→

)・

By Theorem 4.8,limN→ ∞ EIlyAr121=cた >0.■ ol■■this,we mv expeCt thtt the sequencev}N COnК rges in distribution.We could not proК this yet.However,considering the{ン うL2̲lilnit,、 ve get the folloM″ ing result.
jη L2(2,λαS nο れmtt pο れι
Theorem 4。 13.{yyV}N=1,2,… ん).

Lemma 4.14.乃 rf″ Cご rcN,
2■

爵吼 ElyAfyA7]=0・
Pη げ Similarly as in showing the equality(4.15),by uSing Lemma 4.3 and Lemma 4.4
again,、 ve ha■ 7e

ЦttЪ 切 軍Ъ刷 =鳳 EK忍 バo伴 弓 評 二(凛 μO絆×墜饗鰐 ギ ))

些 生―型 二半 ))￨EI(堕学型 ―
=鳳 ΣД
→
嵐
→
■μ)
し,υ くび

×
―
ギ 月
(
=J吼
の
し
Σ響響 し
=ゑ 器 助,M仏 →Jf・ f,N頓

=2Ψ ″
<→
工
←弁
)'
寡
、
vhere

動

,Ⅳ

O‐ ((M modめ ∧O modη り ←―些 型 堕 半 堕 些 金)

is a bounded functiOn. It is easy to see thatOく If71f,N(2)く HN(η ), ∀η ∈ N・Thus,

0く E陥 玲 ]=χ

▼幼Ⅳ v幼こΨ輸'NO寡 ← ― 弁 )―くνⅣこΨ輸0嘉 ←紛v幼v瀦

4.4. Mean square convergence rate

85

―
=脇 蒜こ
Ψ輸0寡 ←
紛=f基:E‖ 玲
向→O as χ→∞.

The lemma is proved.

□Pη げ げ 動 θθκ鶴 4.13.For O<Ⅳ <M,we consider
EI11lY― ンら
v2]=E[ン lv12]+EIン らv21̲2E[ン らvy/vI.
From Lellllna 4.14 and Theoren1 4.8,it fol10ws that

EI聯 ―玲 向 =E]玲 句+α >Q>0.

靭吼This implies that{1lv}Ⅳ has no limit point in Z2(2,入 ).The theorem is proved.

□

Rcmα

0

7・ A・

4.15.Since{￨lyArl12}N iS bOunded,the sequence of probability measures{λ1気 Fl}N On R iS tight.Therefore,for any subsequence{場{N:}such that{入

4.4

o ll勇

1}′

C°

n、rerges weaklyЪ

}′

there exists a subsubsequenceor{16Y;}′ COnverges in distribution.

ⅣIean square convergence rate

χ(2)is multiplicative and X(2)∈ D9fOr all lく 9<∞ ,■ (χ (η ))=χ(χ )fOr all lく 9<∞ .In particular,場 (χ (2))=X(")・ Property(i)of■ (SectiOn 3.2)
Recall that since

ilnplies that for each Ar∈N,乃鰤0‑島 )=鰤 0‑轟)・Thus,by the isollnetric of島 ,lVe get the following result.
Len■ lna 4。 16.Fθ r cα c71 Ar∈ N′

鳳 舟Ё (S・ 0島 )2=EI(鈴 ―島 )21.

四The convergence(4.22)together with the estimate in Theorem 4.8 gives us the estimateof the mean square convergence rate,namely;

Corollary 4.17.̲4s Ar̲→ oo,

)0‑轟
爵
吼券
(Nけ
Ё
υんθtt θん ,stん θcθ nstα ηιづη Tllθ θ 鶴 4.8.

))2〜

QⅣ 7亀

Bibliography[1] A. Axer, Uber einige Grenzwertsd,tze, S.-8. Math.-Natur. K1. Akad. Wi,ss. Wi,en( 2a), t2O:1.253-1298, 191 1.

[2] R.C. Baker and J. Pirftz, The distribution of squarefree numbers, Acta Arith., 46:73
79, 1985.
[3] A.S. Besicovitch and H. Bohr, Almost periodicity and general trigonometric series,
Acta Math., 57 :203-292, 193I.

[4] P. Billingsley, Conuergence of probab'il'ity measures, John Wiley & Sons Inc., New
York-London-Sydney, 1968.

[5] G. Brown and W. Moran, Products of random variables and Kakutani's criterion fororthogonality of product measures, J. London Math. Soc. (2),10:401-405, 1975.
[6] L. Carleson, On convergence and growth of partial sums of Fourier sefies, Acta
Math,., 116:135 157, 1966.
f7] I.P. Cornfeld, S.V. Fomin, and Ya.G. Sinai, Ergod,ic theory, Springer-Verlag, New
York,1982.
[B] J.L. Doob, Stochastr,c processes) John Wiley & Sons, Inc., New York; Chapman &Hall, Limited, London, 1953.

t9] R. Durrett, Probabi,li,ty: theory and eramples, Cambridge Series in Statistical andProbabilistic Mathematics, Cambridge University Press, Cambridge, fourth edition,2010.

f10] Duy, T.K., On the distribution of k-th power free integers, Osaka J. Math.,48 (4) :1027 -I0 45, 2017.

[11] Duy, T.K. and Takanobu, S., On the distribution of k-th power free integers, II,
Osaka J. Math. (accepted)[12] Duy, T.K., Limit-periodic arithmetical functions and the ring of finite integral adeles,
Li,th. Math. J., 5L(4):486-506, 2077.

[13] Duy, T.K., Remarks on value distributions of general Dirichlet series, in Funct'ions
i:n Number Theory and Thei,r Probabi,li,sti,c Aspects-Kyoto 2010. (accepted)[14] Duy, T.K., Carleson's theorem for general Dirichlet series, in Analyti,c and probabi,li,sti,c methods 'in number theory. Proceedi.ngs of the Fi,fth Internat'ional Conference'in honour of J. Kubi,li,us, Palanga 2011. (accepted)[15] P.D.T.A. Elliott, A mean-value theorem for multiplicative functions, Proc. London
Math. Soc. (3),31:418-438, 1975.
87

Bibliography[16] C. Fefferman, On the convergence of multiple Fourier series, Bull. Amer. Math. Soc.,
77:744-745, 797I.
[17] E. Folner, On the dual spaces of the Besicovitch almost periodic spaces, Danske Vt d,.
Selsk. Mat.-Fys. Medd.,29.2T pp., 1954.

[1S] i. Genis and A. Laurindikas, Value distribution of general Dirichlet series. IV, Lith.

Math. J., 43(3):281 294,2003.
[19] J. Genys, D. Siaudiunas, and A. Laurindikas, Value distribution of general Dirichlet
series. VII, Lith. Math. J., 46(2):755-762,2006.
I20] S.W. Graham, The distribution of squarefree numbers, J. Lond,on Math. Soc. (2),24:54 64, 1981.

[21] G.H' Hardy and E.M. Wright, An i,ntroduct'ion to the theory of numbers, Oxford
University Press, Oxford, sixth edition, 2008. Revised bv D.R. Heath-Brown and
J.H. Silverman, With a foreword by Andrew Wiles

l22l H. Hedenmalm and E. Saksman, Carleson's convergence theorem for Diriclet series,
Pacifi,c J. Math., 208:85-109, 2003.
[23] H. Heyer, Probabili,ty measures on locally compact groups, Springer-Verlag, Berlin,
7977.
f24] R.A. Hunt, On the convergence of Fourier series, in Orthogonal Erpanszons and theirCont'inuous Analogues (Proc. Conf., Edwardsui,Ile, lil., lgo7), pp. 235 2b5, Southern
Illinois Univ. Press, Carbondale, Ill., 1968.
[25] K.-H. Indlekofer, New approach to probabilistic number theory-compactifications
and integration, in S. Akiyama, K. Matsumoto, L. Murata, and H. Sugita (Eds.),
Probabi'li'ty and Number Theory-Kanazaua 2005, Adv. Stud. Pure Math., 49, Math.
Soc. Japan, Tokyo, 2007, pp.133 170.
[26] Y. Katznelson i l" introd"ucti;on to harmon'ic analys'is, Cambridge Mathematical
Library, Cambridge University Press, Cambridge, third edition, 2004.
[27) J. Knopfmacher, Fourier analysis of arithmetical functions, Ann. Mat. Pura Appt.
Q ), 1,O9:I77 -207, 1.97 6.
[28] Z. Krizhyus, Limit-periodic arithmetical functions, L'itousk. Mat.,9b., 25(3):93 103,
1985. English translation: Lithuani,an Math. J. 25(3):2a3-250, 1985.
[29] H. Kubota and H. Sugita, Probabilistic proof of limit theorems in number theory by. means of adeles, Kyushu J. Math.,56:391 404,2002.
[30] L. Kuipers and H. Niederreiter, Uniform di,strzbution of sequence.s, Pure and Applied
Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney,
1974.

[31] A. Laurindikas, Value-distribution of general Dirichlet series, in Proceedtngs of the
Seuenth Internati;onal Vi.lnius Conference, V'ilnius, 1998, pp. 405-474, TEV, Vilnius,
1999.

[32) A. Laurindikas, Va]ue distribution of general Dirichlet series. II, Lith. Math. J.,
41(4):351 360, 2001.

Bibliography89[33] A. Laurincikas, Limit theorems for general Dirichlet series) Theorg Stoch. Process.,8(3 4):256 268,2003.
[34] A. Laurincikas, Value distribution of general Dirichlet series. VI, Nonlinear Anal.
Model. Control, 10(3):235 246, 2005.
f35] A. Laurincikas, W. Schwarz, arrd J. Steuding, Value distribution of general Dirichlet
series. III, in Proc. III Palanga Conf., TEV, Vi,lni,us, Analytic and probabilistic
methods in number theory, pp. 137-156, 2002.
f36] J.-L. Mauclaire, Suites limite-p6riodiques et th6orie des nombres. I, Proc. Japan
Acad. Ser. A Math. Sci.,56(4):180 182, 1980.
f37] J.-L. Mauclaire, Int4gratr,on et th6.ori,e des nombres, Travaux en Cours. [Works in
Progress] , Hermann, Paris, 1986.
l3B] H.L. Montgomery and R.C. Vaughan, The distribution of squarefree numbers, inRecent progress i,n analytzc number theory, Vol. 1 (Durham, 1979), pp.247 256, Academic Press, London-New York, 1981.
[39] E.V. Novoselov, A new method in probabilistic number theory, Izu. Akad,. NaukSSSR Ser. Mat., 28:307 364, 7964. English translation: Amer. Math. Soc. Transl.
52:277-275, 1966.
[40] F. Pappalardi, A survey on k-freeness, Number theory, 71 88, Ramanujan Math.
Soc. Lect. Notes Ser., 1, Ramanujan Math. Soc., Mysore,2005.
[41] K.R. Parthasarathy, Probabi,li,ty measures on metric spaces) Probability and Mathematical Statistics, No. 3, Academic Press Inc., New York, 1967.
142] W. Rudin, Fourier analysi,s on groups) Interscience Tracts in Pure and Applied

Mathematics, No. 12, Interscience Publishers (a division of John Wiley and Sons),
New York-London, 1962.
[43] W. Schwarz and J. Spilker, Ari,thmeti,cal functi,ons. An i,ntrod,uct'ion to elementary andanalyti,c properti,es of ari,thmeti,c functi,ons and to some of thezr almost-peri,odi,c propert'ies, Volume 184 of London Mathemati,cal Soci,ety Lecture Note Serr,es, Cambridge
University Press, Cambridge, 1994.

l44l H. Sugita and S. Takanobu, The probability of two integers to be co-prime, revisitedon the behavior of ClT-scaling limit, Osaka J. Math., O(4):945 976,2003.
f45] E.C. Titchmarsh, The theory of functions, Oxford University Press, second edition,
1939.

[46] P. Walters, An i,ntroducti,on to ergodi.c theory, Volume 79 of Graduate Terts i,n Mathemat'ics, Springer-Verlag, New York, 1982.

IndexAdditive function, 65
Besicovitch q-norm, I5, 47
Besicovitch almost periodic function, 15
Besicovitch limit-periodic arithmetical func-tion, 47
Bochner-Fej6r polynomial, 17
Bochner-Fej6r sequence, 17

for functions in flq(f), {^-})' 25
multi-dimensional maximal inequality' 25
Mean valueof Besicovitch almost periodic function,

Ib,20,22of Besicovitch limit-periodic arithmetical function, 56
Multiplicative function, 65, 66

Carleson's theorem / the Carleson-Hunt the- perron's formula, 35

orem

for Besicovitch almost periodic function,29
for classical Fourier series, 24
Characteristic function, 2
Conditional expectation, B
Continuity theorem, 4Dirichlet sequence, 16, 21, 33
Doob's martingale convergence theorem, 11
Ergodic theorem, 5, 7
Fourier expansion, 48, 62
Fourier exponents, 15
Fourier seriesof Besicovitch almost periodic function'
75, 17, 20
of classical Fourier series, 24of limit-periodic arithmetical function,
48

Fourier transform, 4, 22
General Dirichlet series, 33

L6vy's continuity theorem, 3Limit distributionof Besicovitch almost periodic function,

15,20,22of Besicovitch limit-periodic arithmetical function, 47,56
Mapping theorem, 1
I\{aximal inequality
for classical Fourier series. 24

Portmanteau theorem, 1
Ramanujan coefficients, 63