Šiauliai Mathematical Seminar
1(9), 2006, 61–83

AN INTRODUCTION
TO THE VALUE-DISTRIBUTION THEORYOF ZETA-FUNCTIONS
Kohji MATSUMOTO
Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan;
e-mail: kohjimat@math.nagoya-u.ac.jp

Abstract. This is an expository survey on the value-distribution theory ofzeta-functions, mainly of the Riemann zeta-function. We discuss densenessresults, the universality, and limit theorems both in the complex plane andin function spaces.
Key words and phrases: density, Dirichlet series, limit theorem, Riemannzeta-function, universality.
Mathematics Subject Classification: 11M06, 11M38, 11M41.

1.

Basic properties of the Riemann zeta-function

The Riemann zeta-function is defined by the infinite series
ζ(s) =

∞
X

n−s ,

(1.1)

n=1

where s = σ + it is a complex variable. This series is convergent absolutely
in the half-plane D0 = {σ = Re s > 1}, uniformly in any compact subset
of D0 , hence is holomorphic there. Moreover, as was discovered by L. Euler(1737), ζ(s) has the infinite product expression
ζ(s) =∞
Y¡¢−1
1 − p−s
n

(1.2)

n=1

(the Euler product) in D0 , where pn denotes the nth prime number. Consequently ζ(s) 6= 0 in D0 .

62Kohji Matsumoto

The meromorphic continuation of ζ(s) was proved by B. Riemann (1859).
He proved that ζ(s) is holomorphic in the whole complex plane C except forthe only simple pole at s = 1. Riemann also proved the functional equation
µ
¶
³s´
1−s
π −s/2 Γζ(s) = π −(1−s)/2 Γ
ζ(1 − s)(1.3)
2
2
predicted by Euler, by which we can easily reduce the study of ζ(s) in theregion σ < 0 to that in D0 . In particular, we find that ζ(−2k) = 0 for any
k ∈ N (where N is the set of positive integers), and those are the only zeros
in the region σ < 0.
Riemann noticed that the behaviour of ζ(s) in the remaining region 0 6
σ 6 1 (the critical strip), especially the distribution of zeros of ζ(s) in thisstrip, is closely connected to the behaviour of prime numbers. Therefore
the study of ζ(s) in this strip is very important in prime number theory.
In particular, Riemann wrote that it is quite plausible that all the zeros inthis strip are on the line σ = 1/2 (the Riemann hypothesis). However, still
now, no one knows whether the Riemann hypothesis is true or not. Generally
speaking, the study of the behaviour of ζ(s) in this strip is extremely difficult.
Many theorems on the zeros of ζ(s) in this strip are now known, but ourknowledge is still far from the real understanding of ζ(s).
By function theory we can say that we can understand ζ(s) completely if
we understand its zeros and poles completely. This implies that the properties of non-zero values of ζ(s) includes the information of zeros, that is, “whatkind of distribution of zeros may give such properties of non-zero values?”
This gives a motivation of the study of non-zero values of ζ(s). The purposeof the present article is to survey the theory of distribution of non-zero valuesof ζ(s) and more general zeta-functions.
We may expect that, by studying general distribution properties of ζ(s),we can arrive at the global understanding of ζ(s) (which may not be arrived
at if we watch only the zeros). Moreover, the history of the distribution theory of non-zero values shows that the theory is connected with various otherbranches of mathematics, such as probability theory, ergodic theory, functional analysis, and the theory of almost periodic functions. Furthermore,
several unexpected applications of theorems on the value-distribution theory
have been discovered recently. Therefore we can say that the distribution
theory of non-zero values is a rather rich and fruitful area.
This article is an extended written version of the author’s lecture at the
Colloquium on the value-distribution of Dirichlet series, held at Departamento de Matemáticas, Universidad Autónoma de Madrid, Spain, DecemberAn introduction to the value-distribution theory of zeta-functions

63

2005. The author would like to thank Professors Jörn Steuding and Rasa(Šleževičienė-)Steuding for the invitation and hospitality. The author would
also like to thank Professor Antanas Laurinčikas for his suggestion of submitting this written version to the present journal published in Lithuania,
where the value-distribution theory is very actively studied.

2.

The denseness theorem of BohrThe study of the distribution of non-zero values of ζ(s) was initiated by
H. Bohr in 1910s. Because of the functional equation (1.3), it is not strange
to restrict our consideration to the region σ > 1/2. If σ > 1, it is clear that
|ζ(σ + it)| 6 ζ(σ) for any t ∈ R (where R is the set of real numbers), thatis, the orbit ζ(σ + it) is included in a compact subset of C. However, thesituation is different when σ 6 1.
Theorem 1 (Bohr-Courant, [8]). For any σ satisfying 1/2 < σ 6 1, the set{ζ(σ + it) | t ∈ R} is dense in C.
Theorem 2 (Bohr, [7]). For any σ satisfying 1/2 < σ 6 1, the set {log ζ(σ +it) | t ∈ R} is dense in C.
Remark. Since we do not assume the Riemann hypothesis, we cannot
exclude the possibility of existence of zeros of ζ(s) in the strip 1/2 < σ < 1.
Hence, in Theorem 2, we have to decide the branch of the logarithm. Whenσ > 1 we define∞X
log ζ(s) = −Log(1 − p−s
n ),
n=1

where Log denotes the branch of the logarithm which is real on the positive
real axis. Next, in the case 1/2 < σ 6 1, denote by P all (possible) poles
and the zeros of ζ(s) in this strip, and exclude[{σ + it0 | 1/2 < σ 6 σ0 }s=σ0 +it0 ∈Pfrom our consideration. When σ + it is not in the above set, we define thevalue of log ζ(σ +it) by the analytic continuation from the region σ > 1 along
the horizontal line segment.
Actually Theorem 1 is a simple consequence of Theorem 2. In fact, let abe any non-zero complex number. For any ε > 0, Theorem 2 says that we

Kohji Matsumoto

64

can find a t ∈ R such that| log ζ(σ + it) − log a| <

ε
.
2|a|

Hence
|ζ(σ + it) − a| = |a| · |elog ζ(σ+it)−log a − 1|
6 2|a| · | log ζ(σ + it) − log a| < ε,which implies Theorem 1.
Now we briefly sketch how to prove Theorem 2. Let N ∈ N, and consider
the finite truncation
N
Y
¡
¢−1
ζN (s) =1 − p−s
n

(2.1)

n=1

of the Euler product (1.2). Thenlog ζN (σ + it) = −
=−

N
X
n=1
N
X

¡
¢
Log 1 − pn−σ−it

(2.2)

³
´
−it log pn
Log 1 − p−σ
e
.
n

n=1

Define the mapping SN: [0, 1)N → C by

SN (θ1 , . . . , θN ) = −

N
X

³
´
2πiθn
Log 1 − p−σ
e
.
n

(2.3)

n=1

Then from (2.2) and (2.3) we have
¾
½
¾¶
µ½
t
t
,
log ζN (σ + it) = SN
− log p1 , . . . , − log pN
2π
2π

(2.4)

where {x} denotes the fractional part of x.
¡¢2πiθn on the rightWhen θn moves from 0 to 1, each term Log 1 − p−σn ehand side of (2.3) describes a closed convex curve. Therefore
UN = {SN (θ1 , . . . , θN ) 0 6 θn < 1 (1 6 n 6 N )}is a “geometric sum” of convex curves. Since 1/2 < σ 6 1, the series
is divergent. By using this fact, we can show that[
UN = C.
NP −σpnAn introduction to the value-distribution theory of zeta-functions

65

Therefore, for any a ∈ C, we can find a sufficiently large N and θn ∈ [0, 1)(1 6 n 6 N ) for which
SN (θ1 , . . . , θN ) = a(2.5)

holds.
An arithmetic key lemma in the proof of Theorem 2 is the following
approximation lemma of L. Kronecker. Let a1 , . . . , aN ∈ R, linearly independent over the rational number field Q, and let b1 , . . . , bN ∈ R. Then
Kronecker’s lemma asserts that, for any ε > 0, we can find a t > 0 andrational integers m1 , . . . , mN for which|tan − bn − mn | < ε (1 6 n 6 N )(2.6)

holds. Since log p1 , . . . , log pN are linearly independent over Q (which is animmediate consequence of uniqueness of the decomposition of integers into
prime factors), we can apply Kronecker’s lemma to an = − log pn /2π andbn = θn to obtain that
¯
¯
¯ t
¯
¯− log pn − θn − mn ¯ < ε (1 6 n 6 N )
(2.7)
¯ 2π
¯
with some t > 0 and m1 , . . . , mN . Combining (2.4), (2.5) and (2.7) we findthat, under a suitable choice of t, | log ζN (σ + it) − a| can be arbitrarily small.
The only remaining task is to replace log ζN (σ + it) by log ζ(σ + it). Thiscan be achieved by using the following mean value result. Let η > 0. Then,for any sufficiently large N = N (η) and T = T (N, η), one has
¯2
ZT ¯
¯
¯ ζ(σ + it)¯¯¯ ζN (σ + it) − 1¯ dt 6 ηT.

(2.8)

1

This can be shown by using a general mean value theorem of F. Carlson, andthe (classically known) mean square result
ZT|ζ(σ + it)|2 dt = O(T ) (σ > 1/2).

(2.9)

1

Inequality (2.8) implies that the value ζN (σ + it) is not far from ζ(σ + it) for
almost all t. Therefore we obtain the assertion of Theorem 2.
It is to be noted that mean value estimates such as (2.9) are usually
essentially used in the proof of value-distribution theorems.

Kohji Matsumoto

66

3.

Voronin’s multidimensional denseness

The multidimensional generalization of Bohr’s denseness theorem wasproved by S. M. Voronin more than half-century later. Let
D1 = {s ∈ C | 1/2 < Re s 6 1},

D = {s ∈ C | 1/2 < Re s < 1}.

His results can be stated as follows.
Theorem 3 (Voronin, [71]). Let m ∈ N, h > 0.
(i) For any s ∈ D1 , the set{(ζ(s + inh), ζ 0 (s + inh), . . . , ζ (m−1) (s + inh)) | n ∈ N}
is dense in Cm .
(ii) For any s1 , . . . , sm ∈ D1 (distinct from each other), the set{(ζ(s1 + inh), ζ(s2 + inh), . . . , ζ(sm + inh)) | n ∈ N}
is dense in Cm .
The following generalization of Theorem 1 is an immediate corollary of
the assertion (i) of the above theorem.
Theorem 4. For any σ satisfying 1/2 < σ 6 1, the set{(ζ(σ + it), ζ 0 (σ + it), . . . , ζ (m−1) (σ + it)) | t ∈ R}

(3.1)

dense in Cm .
At first glance, this is just a simple generalization of Theorem 1. Actually,
however, this result suggests a rather surprising fact that “any holomorphic
function can be approximated by ζ(s)”.
Let s0 = σ0 + it0 be any point in D. Let U = {s ∈ C | |s − s0 | 6 a} be adisc, U ⊂ D, and f (s) be a function holomorphic on U . Thenf (s) =∞Xf (k) (s0 )k=0k!

(s − s0 )kfor s ∈ U . By Cauchy’s estimate of the coefficients we have¯¯¯ f (k) (s )¯¯0k¯(s − s0 ) ¯ 6 M a−k |s − s0 |k
¯
¯
¯ k!

(3.2)

(3.3)

An introduction to the value-distribution theory of zeta-functions

67

for any k, where M = sup{|f (s)| | s ∈ U }. Let
V0 = {s ∈ C | |s − s0 | 6 δ0 a} ⊂ U
where 0 < δ0 < 1. Then the right-hand side of (3.3) is 6 M δ0k if s ∈ V0 .
Therefore, for any ε > 0, we can find a sufficiently large m = m(M, ε, δ0 ) ∈ N
for which
¯
¯
m−1
¯
¯
X f (k) (s0 )¯¯(s − s0 )k ¯ < ε(3.4)¯f (s) −
¯
¯
k!
k=0

holds for s ∈ V0 .
On the other hand, Theorem 4 implies that we can find a t1 ∈ R forwhich|ζ (k) (σ0 + it1 ) − f (k) (s0 )| < εe−δ0 a(0 6 k 6 m − 1)

(3.5)

holds. Hence
¯m−1
¯
m−1
¯ X f (k) (s )¯(k) (σ + it )
X
ζ
¯
¯
0
0
1
(s − s0 )k −(s − s0 )k ¯
¯
¯
¯
k!
k!
k=0

ε

6 δ a
e0

m−1
X
k=0

k=0

(δ0 a)k
k!

<ε

(3.6)

for s ∈ V0 . Put τ = t1 − t0 . Then σ0 + it1 = s0 + iτ . The Taylor expansionof ζ(s + iτ ) as a function in s at s = s0 isζ(s + iτ ) =∞Xζ (k) (s0 + iτ )
k=0

k!

k

(s − s0 ) =∞Xζ (k) (σ0 + it1 )k=0k!

(s − s0 )k . (3.7)

Let Vδ = {s ∈ C | |s − s0 | 6 δa}, where 0 < δ < δ0 . Again using Cauchy’s
estimate, we find that¯¯¯ ζ (k) (σ + it )
¯
¯
0
1
k¯
k
(s
−
s
)
¯
0 ¯ 6 A(τ )δ
¯¯k!
for s ∈ Vδ , where A(τ ) = sup{|ζ(s + iτ )| | s ∈ U }. Hence, if we chooseδ = δ(m, ε, τ ) sufficiently small, we have
¯
¯
m−1
¯
¯
X ζ (k) (σ0 + it1 )
¯
k¯
(3.8)
ζ(s
+
iτ
)
−
(s
−
s
)
¯
0 ¯<ε
¯
¯
k!
k=0

Kohji Matsumoto68

for s ∈ Vδ . From (3.4), (3.6) and (3.8) we obtainsup |ζ(s + iτ ) − f (s)| < 3ε.

(3.9)

s∈Vδ

This implies that f (s) can be approximated by (a certain translation of) ζ(s)uniformly in the disc Vδ . This kind of property is called universality of ζ(s).
The above argument of deducing (3.9) from Theorem 4 is sketched in
the author’s textbook [56], written in Japanese. An unsatisfactory point of(3.9) is that it holds only in a small disc Vδ , and moreover, the radius δaof Vδ depends on τ . This point is overcome in the stronger version of theuniversality due to Voronin, which will be discussed in the next section.

4.

The universality

Denote by µ1 {A} the 1-dimensional Lebesgue measure of the set A. We
use the notation1νT (· · · ) = µ1 {τ ∈ [0, T ] | · · · }T
for T > 0, where in place of dots we write a certain condition satisfied by τ .
Let K be a compact subset of D, and F(K) be the family of functions which
are non-vanishing, continuous on K and holomorphic in the interior of K.
Then
Theorem 5 (The universality theorem of Voronin). For any compact subset
K of D with connected complement, we have½¾
lim inf νT sup |ζ(s + iτ ) − f (s)| < ε > 0
(4.1)
T →∞

s∈K

for any f ∈ F(K) and any ε > 0.
This formulation of the universality theorem is due to Reich [65]. Voronin’s original statement in [72] is restricted to the case when K is a disc inD, but the above form of the universality can be deduced by his argument.
It is a natural question how to generalize this remarkable result to moregeneral zeta and L-functions. Let χ be a Dirichlet character mod q, andL(s, χ) the associated Dirichlet L-function. Voronin [72] already mentionedthat his result can be generalized to L(s, χ). More strongly, the following“joint universality theorem” holds (see Voronin [73], Gonek [15], and Bagchi[2], [3]).

An introduction to the value-distribution theory of zeta-functions

69

Theorem 6. Let K1 , . . . , Km be compact subsets of D with connected complements, and fj ∈ F(Kj ) (1 6 j 6 m). Let χ1 , . . . , χm be pairwise nonequivalent Dirichlet characters. Then we have()
lim inf νTT →∞max sup |L(s + iτ, χj ) − fj (s)| < ε

16j 6m s∈Kj

>0

(4.2)

for any ε > 0.
In the proof of Theorem 2, we used the fact that log p1 , . . . , log pN are
linearly independent over Q. The same fact is also used in the proof of
Theorem 5. To prove Theorem 6, the orthogonality property of Dirichlet
characters is also applied.
Let ζ(s, α) be the Hurwitz zeta-function with parameter α (0 < α 6 1).
If α is rational or transcendental, then ζ(s, α) has the universality property,
that is, the inequality of the form (4.1) with replacing ζ(s+iτ ) by ζ(s+iτ, α)holds (Bagchi [2], Gonek [15]). In this case the non-vanishing condition forf (s) is not necessary. This is because we do not use the Euler productexpansion in the proof; actually Hurwitz zeta-function in general has no Euler
product. Instead, when α is rational, we write ζ(s, α) as a linear combinationof Dirichlet L-functions and apply Theorem 6. When α is transcendental,
we use the fact that log(n + α) (n ∈ N) are linearly independent over Q.
When α is algebraic irrational, to prove the universality of ζ(s, α) is an open
problem.
In general, we call a Dirichlet series Z(s) strongly universal if (4.1) withreplacing ζ(s + iτ ) by Z(s + iτ ) holds. Now it is known that various zeta andL-functions are strongly universal; Dedekind zeta-functions of algebraic number fields (Voronin [74], Gonek [15], Reich [66], [67]), Lerch zeta-functions(Laurinčikas [30], [31], see also Laurinčikas and Garunkštis [43]), Dirichlet series attached to finite Abelian groups (Laurinčikas [37], [39]), Rankin-Selberg
L-functions (the author [54]), etc.
The universality in some general setting has also been studied. This direction was cultivated by Reich [65], and then, pursued further by Laurinčikas
([20], [21], [22], [23], [24], [32]). Joint universality theorems in general setting
are discussed in Laurinčikas [25], [33]. See also the next section.
An important problem is to find some quantitative version of universality
theorems. The first attempt to this direction is due to Good [14]. Recently,
this problem has been studied by Laurinčikas [35], Steuding [69], and Garunkštis [11]. Steuding considered the upper density (defined by replacing
lim inf by lim sup on the left-hand side of (4.1)) and obtained an upper boundKohji Matsumoto70

of it, while Garunkštis, inspired by the work of Good [14], obtained a lower
bound of the lower density when K is sufficiently small.
There are three textbooks ([19], [27], [43]) in which the universality istreated. Also there are several survey papers, such as Laurinčikas [36], [39],[41], the author [55]. Steuding’s lecture note on this topic will appear soon.

5.

The positive density method

There is a conjecture, due to Yu. V. Linnik and I. A. Ibragimov, whichsays that all functions defined by Dirichlet series, which can be continued
analytically to the left of the abscissa of absolute convergence and satisfy
some natural conditions there, would be strongly universal. In view of thisconjecture, it is desirable to show universality property for more and morezeta and L-functions.
In this direction, one important turning point is the unconditional proof ofthe strong universality of automorphic L-functions attached to cusp forms for
SL(2, Z), due to Laurinčikas and the author [45]. Let F (z) be a holomorphic
normalized Hecke-eigen cusp form of weight κ for SL(2, Z), and c(n) the nth
Fourier coefficient of F (z). Then the associated Dirichlet series
L(s, F ) =∞X

c(n)n−sn=1

is convergent absolutely for σ > (κ + 1)/2, and can be continued holomorphically to the whole plane. Let
D(κ) = {s ∈ C | κ/2 < σ < (κ + 1)/2}.
Then
Theorem 7 ([45]). For any compact subset K of D(κ) with connected complement, we have½¾
lim inf νT sup |L(s + iτ, F ) − f (s)| < ε > 0
(5.1)
T →∞

s∈K

for any f ∈ F(K) and any ε > 0.
The behaviour of c(p), when p runs over the set of primes, is very complicated. This situation causes a trouble when one try to prove the above
theorem analogously to Theorem 5. However, by a certain mean value resulton c(p) we can deduce that the set of primes, for which |c(p)| is not so small,An introduction to the value-distribution theory of zeta-functions

71

is of positive density. By using this fact, we can go through the obstacle toobtain Theorem 7.
This idea (“the positive density method”) has then been repeatedly usedin the study of universality. In fact, the method in [45] has been generalized
to show the universality of L-functions of new forms in [48], and the joint
universality of twisted automorphic L-functions in [46]. Laurinčikas and theauthor [44] used a subset of positive density, not in the set of primes, but inthe set of positive integers, to prove a joint universality theorem for Lerch
zeta-functions. (Note that the results in [44] are valid under the condition
that α1 , . . . , αn are algebraically independent over Q, which is lacking inthe statements. See [47].) Mishou [57], [58] used certain sets of primes of
positive density to show that Hecke L-functions of algebraic number fields
are strongly universal. The universality of Artin L-functions has been proved
by Bauer [4].
The positive density method has been applied in a more general situation.
Laurinčikas and Šleževičienė [50] applied this method to show a universality
theorem for general zeta-functions with multiplicative coefficients. Šleževičienė [68] generalized this result to the joint case, which has been further
applied to the proof of the universality of Estermann zeta-functions [13].
The universality of zeta-functions belonging to a certain subclass of Selberg class has been shown by Steuding [70]. Laurinčikas, Schwarz and Steuding [49] proved a universality theorem for a certain type of general Dirichlet
series, and its joint version was obtained by Laurinčikas [40], [42].
For any holomorphic cusp form F (z), the Ramanujan-Petersson conjecture |c(p)| 6 2p(κ−1)/2 was proved by P. Deligne, and this was used in [45].
However, the corresponding conjecture has not yet been proved for Maass
forms. Nagoshi [63] tried to prove the universality for L-functions attachedto Maass forms, and he proved it under a certain condition weaker thanthe Ramanujan conjecture. Then in [64], Nagoshi succeeded in proving theuniversality of Maass form L-functions unconditionally, by employing a certain fourth power mean estimate of the coefficients. The readers will noticethat the total number of papers discussing universality of various zeta andL-functions is increasing very rapidly in recent years. It is impossible here
to mention all relevant papers.
The universality property may be regarded as an ergodic property ofzeta-functions in function spaces. Therefore it is natural to understand thisproperty in a formulation of function spaces, and in fact, it is closely connected with functional limit theorems of zeta-functions. To explain this connection, however, it is better to begin with limit theorems on the complex
plane, which goes back to Bohr’s work.

Kohji Matsumoto

72

6.

Limit theorems on the complex plane

Let R be a given rectangle in the complex plane C with the edges parallelto the axes, and define
V (T ; σ, R) = µ1 {t ∈ [0, T ] | log ζ(σ + it) ∈ R},
where T > 0. As a refinement of his denseness theorems (Theorems 1 and
2), Bohr proved the following result in a collaborated work with Jessen:
Theorem 8 (Bohr-Jessen, [9]). For any σ > 1/2, there exists the limit
W (σ, R) = lim

1

T →∞ T

V (T ; σ, R).

(6.1)

This limit W (σ, R) may be regarded as the “probability” of how many
values of log ζ(s) on the line Re s = σ belong to R. The proof of this theorem
given in [9] is based on the same idea as in the proof of Theorem 2 sketchedin Section 2. In particular, the fact that each term on the right-hand sideof (2.3) is convex is important in their proof. However, for more generalzeta-functions, the corresponding term is not always convex. The author[51], [52], [53] presented alternative proofs which are free from convexity,
hence succeeded in proving an analogous result for a certain general class ofzeta-functions defined by Euler products.
It is more convenient to formulate the above type of limit theorems in
the frame of modern probability theory.
We recall some basic notion of probability theory. The triple (S, F, P ),where S is an non-empty set, F is a σ-field consisting of some subsets of S,
and P is a probability measure on S, is called a probability space. Hereafter
we consider the case that S is a metric space and F = B(S) is the family of
all Borel subsets (that is the σ-field generated by all open subsets) of S. We
call a set A ∈ B(S) a continuity set with respect to P if P (∂A) = 0, where∂A is the boundary of A. Let Pn (n ∈ N) and P be probability measures on(S, B(S)). Then the following three assertions are equivalent:(i) For any real bounded continuous function f on S,ZZ
f dPn → f dP (n → ∞),
(6.2)
S

S

(ii) For any continuity set A with respect to P ,lim Pn (A) = P (A),

n→∞

(6.3)

An introduction to the value-distribution theory of zeta-functions

73(iii) For any open set G,
lim inf Pn (G) > P (G).
n→∞

(6.4)

When these three assertions are valid, we say that Pn converges weakly
to P , and we write Pn ⇒ P . This is the standard definition of weak convergence, but for our purpose, it is necessary to consider the family {PT }with a continuous parameter T > 0. When T → ∞ continuously, the threestatements analogous to the above are also equivalent, hence we can define
the weak convergence PT ⇒ P (see the comment after Theorem 2.3 of [5]).
For any σ > 1/2 and 0 < γ < δ, let
Wγ,δ,σ (A) =1· µ1 {γ < t < δ | log ζ(σ + it) ∈ A}
δ−γ

where A ∈ B(C). Then Wγ,δ,σ is a probability measure on C, and we have
Theorem 9 (Borchsenius-Jessen, [10]). For any σ > 1/2, there exists a
probability measure Wγ,σ on C for which Wγ,δ,σ ⇒ Wγ,σ holds as δ → ∞.
When A is a rectangle R (with the edges parallel to the axes), we canshow that R is a continuity set with respect to the limit measure (see, e. g.,(4.1) of [52]). Hence from (6.3) and the above theorem we havelim W1,T,σ (R) = W1,σ (R),T →∞which implies Theorem 8.
The proof of Borchsenius and Jessen [10] is based on the theory of almost
periodic functions. This theory was also created by Bohr in connection withthe value-distribution theory of zeta-functions, though he did not use thistheory in [9].
In Borchsenius and Jessen [10], an analogue of Theorem 9 for (not log ζ(s)but) ζ(s) has been proved. From such an analogue, we can show the following
corollary. Let T > 0, σ > 1/2, and define
PT,σ (A) =1µ1 {t ∈ [0, T ] | ζ(σ + it) ∈ A}
T

for any A ∈ B(C). Then
Theorem 10. For any σ > 1/2, there exists a probability measure Pσ on C
for which PT,σ ⇒ Pσ holds.

Kohji Matsumoto

74

This is the form stated as Theorem 4.1.1 of Laurinčikas [27]. This kindof limit theorems has now been shown in a very general situation; see Laurinčikas [38]. Discrete analogues of Theorem 10 for another general class ofzeta-functions can be found in Kačinskaitė [17] and Kačinskaitė and Laurinčikas [18].

7.

Limit theorems in function spaces

Let H(D) be the set of all functions holomorphic on D. Introducing the
topology of uniform convergence on compact subsets, we can regard H(D)as a topological space. There exists a family of compact subsets Kn of D(n ∈ N) such that
K1 ⊂ K2 ⊂ · · · ,

D=

∞
[

Kn ,

n=1

and for any compact subset K of D, there exists an n for which K ⊂ Kn
holds. For any f, g ∈ H(D), let
ρn (f, g) = sup |f (s) − g(s)|,s∈Knand defineρ(f, g) =

∞
X

2−n ·

n=1

ρn (f, g).
1 + ρn (f, g)

Then ρ is a metric on H(D), which induces the above topology.
For any A ∈ B(H(D)), definePT∗ (A) =1µ1 {τ ∈ [0, T ] | ζ(s + iτ ) ∈ A}.
T

Then PT∗ is a probability measure on H(D), andTheorem 11 (Bagchi, [2]). There exists a probability measure Q∗ on H(D)such that PT∗ ⇒ Q∗ as T → ∞.
This is clearly a “functional analogue” of Theorem 10. Moreover, Bagchi
discovered an explicit form of the limit measure Q∗ . Denote by γ the unit
circle in the complex plane, and let
Y
Ω=
γp ,
p

An introduction to the value-distribution theory of zeta-functions

75

where p runs over all primes and γp = γ for each p. This is naturally acompact Abelian group, hence there exists the unique Haar measure mH onΩ with mH (Ω) = 1. Then (Ω, B(Ω), mH ) is a probability space. For ω ∈ Ω,
denote by ω(p) the projection of ω to the coordinate space γp . Define¶Yµω(p) −1ζ(s, ω) =
1− s
.
p
p

(7.1)

Let 1/2 < σ0 < 1. Then ζ(σ0 , ω) converges almost surely (with respect tothe measure mH ), and equals to∞Xω(n)n−σ0 ,

(7.2)

n=1

where ω(n) = ω(p1 )α1 · · · ω(pr )αr if n = pα1 1 · · · pαr r . Using the theory of
Dirichlet series we find that the convergence of (7.2) implies the convergenceof the series∞Xω(n)n−s ,

(7.3)

n=1

uniformly on any compact subset of the half-plane σ > σ0 . Since (7.3) equals
to the right-hand side of (7.1), we can conclude that (7.1) converges almost
surely on any compact subset of D, and hence an element of H(D). Thereforeω 7→ ζ(s, ω) is an H(D)-valued random element. Let Qζ be the distributionof this random element, that isQζ (A) = mH (ω ∈ Ω | ζ(s, ω) ∈ A) (A ∈ B(H(D))).
Then
Theorem 12 (Bagchi, [2]). We have Qζ = Q∗ .
From these theorems we can deduce an explicit form of Pσ in Theorem 10.
In fact, let h be a mapping from H(D) to C defined by h(f ) = f (σ) for any
f ∈ H(D). Then PT,σ = PT∗ ◦ h−1 . Since h is continuous, noting Theorem
1.1.16 of [27], from Theorems 11 and 12 we find that Pσ = Q∗ ◦h−1 = Qζ ◦h−1 ,that is,Pσ (A) = mH (ω ∈ Ω | ζ(σ, ω) ∈ A)
for any A ∈ B(C). This argument (suggested by B. Grigelionis) is presented
in Laurinčikas [26], and also in Notes of Chapter 5 of [27].

Kohji Matsumoto

76

Bagchi used his Theorems 11 and 12 to give a new proof of Voronin’s
universality theorem. Here we sketch his argument.
The support Supp(P ) of a probability measure P on a probability space
S is defined bySupp(P ) = {x ∈ S | P (G) > 0 for any neighbourhood G of x}.
Another key of Bagchi’s proof of the universality is the fact that
Supp(Qζ ) = {ϕ ∈ H(D) | ϕ(s) 6= 0 for any s ∈ D, or ϕ ≡ 0}.

(7.4)

This is a consequence of the following “denseness lemma” of Bagchi. Fors ∈ D and ap ∈ γ, put fp (s) = − log(1 − ap p−s ).
Theorem
13 (Bagchi, [2]). The set of all convergent series of the formP
p fp (s) is dense in H(D).
Now, let f (s) ∈ F(K), and assume that f (s) can be continued to aholomorphic function on D, and f (s) 6= 0 for any s ∈ D. Since the set
¯
½
¾
¯
¯
G = g ∈ H(D) ¯ sup |g(s) − f (s)| < εs∈Kis open, Theorems 11, 12 and (6.4) implies that
lim inf PT∗ (G) > Qζ (G).
T →∞

(7.5)

On the other hand, f ∈ Supp(Qζ ) by the assumption and (7.4). Since G isa neighbourhood of f , we have Qζ (G) > 0. From this and (7.5) we have
lim inf PT∗ (G) > 0,T →∞which is exactly (4.1).
In general case, we use a theorem of Mergelyan, which asserts that any
function f ∈ F(K) can be approximated by a certain polynomial uniformly
on K. Then, applying the above argument to that polynomial, we can obtain(4.1).
The details of Bagchi’s proof of (4.1) can be found in Chapter 6 of Laurinčikas [27].
In Sections 4 and 5, we mention many papers in which various generalizations of Voronin’s universality theorem are treated. In most of those papers,

An introduction to the value-distribution theory of zeta-functions

77

the argument is based on Bagchi’s idea. In order to generalize Bagchi’s
method, it is necessary to generalize Theorems 11, 12, and 13. Now, indeed, Theorems 11 and 12 have been generalized to a fairly general class ofzeta-functions which have Euler products introduced by the author [52]; see
Laurinčikas [28], [29].
Moreover, even in the case of Dirichlet series without Euler products, it
is possible to prove an analogue of Theorems 11 and 12 under some suitableconditions. In this case, instead of the above Ω, the basic probability space
is of the form
∞
Y
e=
Ω
γm ,
m=1

where γm = γ for all m ∈ N. The case of Lerch zeta-functions is written
in [43]. For more general Dirichlet series, see Laurinčikas [34] (partly with
Schwarz, Steuding and Genys). Various discrete versions and joint versionsof those limit theorems have also been extensively studied.
On the other hand, in general, Theorem 13 cannot be easily generalized.
This is because the proof of Theorem 13 is based on a deep arithmeticalresult, that is the asymptotic formula
X1
p6x

p

= log log x + a1 + O(exp(−a2plog x),(7.6)

where the sum runs over all primes 6 x, and a1 , a2 are constants with a2 > 0.
It is sometimes very difficult to obtain such an asymptotic formula in thecase of more general zeta-functions. The positive density method explainedin Section 5 is a technique which, in some sense, avoids the role of such an
asymptotic formula in the proof of universality.
The original proof of the universality theorem due to Voronin himself
is also still useful. For example, the argument in Mishou [57] and Nagoshi[63] are inspired by Voronin’s method. Garunkštis [12] pointed out that
it is possible to remove the rearrangement argument from Voronin’s proof,
which sheds some light on the effectivization problem. Voronin’s idea is also
important in Nagoshi’s recent work on the joint universality of automorphic
L-functions.
Finally, it is to be stressed that various interesting applications of universality theorems have already been discovered. Applications to the functional
independence and the distribution of zeros of Dirichlet series were alreadystudied by Voronin himself. Andersson [1] used the universality to disprovea conjecture of Ramachandra. The connections between universality andquantum physics were studied by [6], [16]. Recent series of joint papers ofKohji Matsumoto

78

Mishou and Nagoshi [60], [61], [62], and also Mishou [59], studied universality
properties of L-functions of number fields, and deduced interesting arithmetical consequences, such as some distribution properties of class numbers ofalgebraic fields.
These applications suggest that the importance of limit theorems anduniversality theorems in the theory of zeta-functions will increase more andmore in the near future.

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Received
12 June 2006