SCHEDULED TALKS AND THEIR ABSTRACTS FOR”DIOPHANTINE ANALYSIS AND RELATED FIELDS 2011”
Martin Stein (FHDW Hannover)
Title: Algebraic Independence Results for Reciprocal Sums of Fibonacci and Lucas Numbers
Abstract: Let Fn and Ln denote the Fibonacci and Lucas numbers, respectively.
D. Duverney, Ke. Nishioka, Ku. Nishioka and I. Shiokawa proved that the values of theP∞−2s
Fibonacci zeta function ζF (2s) =are transcendental for any s ∈ N using
n=1 FnNesterenko’s theorem on Ramanujan functions P (q), Q(q), and R(q). They obtained simP−2silar results for the Lucas zeta function ζL (2s) = ∞and some related series. Later,n=1 LnC. Elsner, S. Shimomura and I. Shiokawa found conditions for the algebraic independenceof these series. In my PhD thesis I generalized their approach and treated the following
problem: We investigate all subsets of)
(∞∞∞∞
X 1 X
(−1)n+1 X 1 X (−1)n+1,,,: s1 , s2 , s3 , s4 ∈ N ,34
F 2s1 n=1 Fn2s2 n=1 L2sL2snnn=1n=1 n
and decide on their algebraic independence over Q. Actually this is a special case of amore general theorem for reciprocal sums of binary recurrent sequences.
Carsten Elsner (FHDW Hannover)
Title: New Algebraic Independence Results Based on a Theorem of Y. Nesterenko
Abstract: In their joint work during the last years the authors established a method
how to decide on the algebraic independence of a set {y1 , . . . , yn } when these numbers
are connected with a set {x1 , . . . , xn } of algebraic independent parameters by a systemfi (x1 , . . . , xn , y1 , . . . , yn ) = 0 (i = 1, 2, . . . , n) of rational functions. Constructing algebraic
independent parameters by Nesterenko’s theorem, the authors successfully applied their
method to reciprocal sums of Fibonacci numbers and determined all the algebraic relations
between three q-series belonging to one of the sixteen families of q-series introduced by
Ramanujan.
In this talk we apply the method mentioned above to various sets of numbers. Our
algebraic independence results include the numbers F2k (q)/(q; q)3∞ from Ramanujan’s Lost
Notebook, where q ∈ Q with 0 < |q| < 1 andF2k (q) :=∞Xn−1(−1)(2n − 1)

2k+1 n(n−1)/2q,(q; q)∞ :=n=1∞
Y(1 − q n ) .

n=1r

By a second application we treat the values P (q ), Q(q ), and R(q r ) of the Ramanujan
functions P, Q, and R, for q ∈ Q with 0 < |q| < 1 and r = 1, 2, 3, 5, 7, 10. Finally, weconsider the values given by reciprocal sums of polynomials.
1/7r

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DIOPHANTINE ANALYSIS AND RELATED FIELDS 2011

Nikolay Moshchevitin (Moscow Lomonosow Univ.)
Title: Multi-dimensional Diophantine approximations and their applications
Abstract: In this lecture I suppose to consider the following topics.
1. Khintchine’s singular matrices. In 1926 A. Khintchine ﬁnd principal differences between one-dimensional and multi-dimensional Diophantine approximations. He discovered the phenomenon that in dimension 2 and greater there exist vectors admitting anomaly good approximations by rationals. These singularvectors (and in the case of system of linear forms in Rd -singular matrices) may
admit very unexpected Diophantine properties. One of these properties is the phenomenon of degeneracy of dimension of rational subspaces generated by the best
approximations. Here we shall give a brief review of classical results and recent
results by the author.
2. Diophantine exponents. There are diﬀerent types of multi-dimensional Diophantine exponents which characterize the rate of approximation of a certain linear
subspace in Rd by rational subspaces. A principal result witch gives lower boundfor the ordinary Diophantine exponent in terms of uniform Diophantine exponent(the last may be considered as a measure of singularity) was obtained by V. Jarnik
in 1954. Recently such a result turned out to be important in diﬀerent problems in
Diophantine approximations. M. Laurent showed that in two-dimensional setting
Jarnik’s bound cannot be improved. It is not so in dimensions greater than two.
We shall discuss diﬀerent deﬁnitions, give a brief review on the topic and introduce
new improvements of Jarnik’s inequality
3. Kozlov’s problem in uniform distribution. In 1978 V. Kozlov posed a problem of behavior of the integral of a quasi-periodic function. This problem turned
out to be connected not only to the theory of dynamical systems but mostly toDiophantine approximations. This problem was completely solved by the author
in 1998. In fact the most diﬃculty in this problem occurs due to Khintchine’s
singular matrices. Here we suppose to discuss certain Diophantine aspects of thisresult.
Yohei Tachiya (Keio University)
Title: Algebraic independence of inﬁnite products generated by Fibonacci numbers
Abstract: Let α and β be real algebraic numbers with |α| > 1 and αβ = −1. We deﬁneαn − β nUn =α−βand Vn = αn + β n(n ≥ 0).

√If α = (1 + 5)/2, we have Un = Fn and Vn = Ln (n ≥ 0), where the sequences{Fn }n≥0 and {Ln }n≥0 are Fibonacci numbers and Lucas numbers deﬁned, respectively,
by Fn+2 = Fn+1 + Fn (n ≥ 0), F0 = 0, F1 = 1 and Ln+2 = Ln+1 + Ln (n ≥ 0), L0 = 2,L1 = 1. Let d ≥ 2 be a ﬁxed integer.

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In this talk, we give necessary and suﬃcient conditions for the inﬁnite products¶¶∞ µ∞ µYYaiai
1+(i = 1, . . . , m) or1+
(i = 1, . . . , m)
Udk
Vdk
k=1
k=1
U k 6=−aidV k 6=−aidwith nonzero integers a1 , . . . , am to be algebraically independent over Q. As an application, we obtain the algebraic independece over Q of the numbers¶∞ µYai1+(i = 1, . . . , m)Fdk
k=1
F k 6=−aidfor any ﬁxed integer d ≥ 2 and any nonzero distinct integers a1 , . . . , am . In particular,
QQ∞
two numbers ∞k=2 (1 − 1/F2k ) and
k=1 (1 + 1/F2k ) are algebraically independent over Q.

Takaaki Tanaka (Keio Univ.)
Title: Algebraic independence properties related to certain inﬁnite products and Lambert
seriesAbstract: In this talk we establish algebraic independence of the values of a certain
inﬁnite product as well as its successive derivatives at algebraic numbers, using the fundamental relation between inﬁnite products and Lambert series. The inﬁnite producttreated in this talk has a simpler form than that talked at RIMS in October 2010. The
method used in the proof of algebraic independence is based on the theory of Mahler
functions of several variables, whose introduction will also be included in this talk.
Ryotaro Okazaki (Doshisha Univ.)
Title: Weber’s Class Number Problem and Pohst’s Lower bound for Algebraic³pUnits√ ´¡√¢
Abstracts: Let Vn = Q(cos(2π/2n+2 )). Then, V1 = Q 2 , V2 = Q2+ 2 ,µq¶p√V3 = Q2 + 2 + 2 ,. . . . The speaker proposes to call Vn the n-th Vieté ﬁeld.
The ring On of integers of Vn is Z[2 cos(2π/2n+2 )]. Weber showed On is a PID (principalideal domain) for n = 1, 2 and 3. Then, he asked if On is always a PID. Later, Bauerand Masley showed O4 is also a PID. and Linden showed O5 is also a PID. RecentlyK.Horie initiated a project for aﬃrmative answer to Weber’s problem. K.Horie, M.Horie,Fukuda, Komatsu, Morisawa made signiﬁcant contribution. Fortuntately, the speaker isalso involved in this project.
The class number hn of Vn measures how far is On from being PID. It vanishes (i.e.,hn = 1) if On is a PID. We prove a certain proposition on the prime numbers which
are coprime with every hn (n ≥ 1). The key idea is Pohst’s lower bound on units withconsideration on congruence incorporated.

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DIOPHANTINE ANALYSIS AND RELATED FIELDS 2011

Ryuta Hashimoto (Kagawa Nat. Col. Tech.)
Title: Searching discriminants with large fundamental units via continued fraction expansionsAbstract: We discuss necessary conditions for a discriminant of some quadratic order not
to be large compared with its fundamental unit, in terms of continued fraction expansions.

Hajime Kaneko (Kyoto Univ.)
Title: On the b-ary expansions of algebraic irrational numbers
Abstract: Borel conjectured that all algebraic irrational numbers are normal in any
integral base b. However, very little is known on the digits of base b expansions ofalgebraic irrational numbers. In this talk we consider the numbers of digit changes in theb-ary expansions of algebraic irrational numbers. I will give new lower bounds for thenumbers of digit changes for certainclasses of algebraic irrational numbers. If we havetime, I will also discuss the p-adic expansions of algebraic irrational numbers in the ﬁeld
Qp of p-adic numbers.
Jonathan Sondow (New York)
Title: Some consequences of Schanuel’s conjecture
Abstract: Assuming Schanuel’s conjecture (S), we prove that if z and w are complex
numbers, not 0 or 1, with z w and wz algebraic, then z and w are either both rationalor both transcendental. A corollary is that if (S) is true, then we can ﬁnd four distinct
transcendental positive real numbers x, y, s, t such that the three numbers xy 6= y x andst = ts are all √integers. Another application is that (S) implies the transcendence of the√ √2 2 ii
πznumbers 2, i , and ie . We also prove that if (S) holds and aa = z, where a 6= 0is algebraic and z is irrational, then z is transcendental. This is joint work with DiegoMarques. Our paper is to appear in the East-West J. of Math. A preprint is available at
http://arxiv.org/abs/1010.6216.
Yuri V. Nesterenko (Moscow State Univ.)
Title: On the measure of irrationality of some numbers
Abstract: For any real irrational number α the exponent of irrationality µ(α) is deﬁned
as the supremum of the set of all numbers κ such that inequality¯¯¯¯¯α − p ¯ < q κ¯q¯p
has inﬁnitely many solutions in rational numbers . We know that µ(e) = 2 and µ(α) = 2q
for any algebraic irrational number α by K. Roth in 1954. Unfortunately precise valuesof µ(α) are known only for a small set of numbers. For some classical constants we know

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only upper bounds for the exponent of irrationality. For example the best up today boundµ(π) ≤ 7, 606308 . . .
was proved in 2008 by V.H. Salikhov. The best known inequalitiesµ(ζ(2)) ≤ 5, 441242 . . . ,µ(ζ(3)) ≤ 5, 513890 . . . ,were proved in 1996 and 2001 in joint papers of G. Rhin and C. Viola. In 2009 R. Marcovecchio proposed a new construction of rational approximations to logarithms of rational
numbers and proved inequality(1)µ(ln 2) ≤ 3, 57455390 . . . .

In the talk we discuss the Salikhov’s construction of rational approximations to π anda new construction of rational approximations to ln 2 that give upper bound (1) and issimpler than Marcovecchio’s one.
Yasutsugu Fujita (Nihon Univ.)
Title: Generators for the elliptic curve y 2 = x3 − nx II
Abstract: Let E be an elliptic curve over the rationals Q given by y 2 = x3 − nx witha positive integer n. Mordell’s theorem asserts that the group of rational points on E is
ﬁnitely generated. Our interest is in the generators for its free part. In our recent work,
we gave several inﬁnite families of n for which certain 2 points of inﬁnite order can always
be in a system of generators for the Mordell-Weil group E(Q). In this talk, we will extend
the above work and give those examples of inﬁnite families of n for which the generatorsof rank 3 part for E(Q) can be explicitly described.
Pingzhi Yuan (South China Normal Univ.)
Title: Ko’s method and its application to Diophantine equations
Abstract: In this talk, we will present a survey on Ko’s method and its application toDiophantine equations. We will give the development of the Diophantine equations of theform AX 4 − BY 2 = ±1, 2, 4. Now most equations of the form AX 4 − BY 2 = ±1, 2, 4
have been completely solved by Ko’s method and the method of Thue-Siegel. We canalso improve the above two methods to solve other interesting equations, for example,
we proved that the diophantine equation x2 − (1 + a2 )y 4 = −2a, a > 0 has at mostthree positive integer solutions (x, y); and we proved that: Let a > 1 and D > 1 bepositive integers, and let p be an odd prime which does not divide aD. If (a, D, p) 6=(1, 2, 5), (1, 4, 5), (2, 7, 3), (2, 7, 9), then the diophantine equation ax2 + Dm = pn has at
most two positive integer solutions (x, m, n). Moreover, the diophantine equations x2 +2m = 5n , x2 + 4m = 5n , 2x2 + 7m = 3n and 2x2 + 7m = 9n have precisely three positive
integer solutions (x, m, n).

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DIOPHANTINE ANALYSIS AND RELATED FIELDS 2011

Takafumi Miyazaki (Tokyo Metropolitan Univ.)
Title: Exceptional cases of Terai’s conjecture on Diophantine equations
Abstract: Let p, q, r be positive integers greater than 1, and let a, b, c be relatively prime
positive integers such that ap + bq = cr . Terai’s conjecture states that if (a, b, c) is not anyof the following cases (taking a < b):
1i + 23 = 32 ; i ≥ 2,25 + 72 = 34 ,2p + (2p−2 − 1)2 = (2p−2 + 1)2 ; p ≥ 4(we call these cases exceptional cases), then the only solution of the equationax + b y = c z ,x, y, z ∈ Nis (x, y, z) = (p, q, r). In this talk, we consider the case where q = r = 2 and give someresults related to exceptional cases.
Tomohiro Yamada (Kyoto Univ.)
Title: On diophantine equation x2 = pg + pf + pe + 1
Abstract: We show that the equation x2 = pg + pf + pe + 1 has only ﬁnitely manysolutions in integers x, e, f, g for each prime p under the condition e is even or both off, g are even. We give an explicit bound for such solutions x, e, f , g.
Shigeki Akiyama (Niigata Univ.)
Title: On the spectrum of {−1, 0, 1} polynomials
Abstract: Given q > 1, consider the set¯( n)¯X¯Y (q) =ai q i ¯ ai ∈ {−1, 0, 1}, n = 0, 1, . . .
¯i=0which is sometimes called the spectrum of q. Originated by P. Erdős and his collaborators,
many authors studied its topology in relation to the expansion in non-integer base q,
Diophantine property of Y (q), and characterization of singular Bernoulli convolution.
Our main question is for which q, the spectrum Y (q) is dense, discrete, uniformly discretein R. We review the situation and explain our recent best possible result on discreteness:
Y (q) is closed and discrete if and only if q is a Pisot number or q ≥ 2
obtained together with V. Komornik. This result gives many consequences for densenessand uniformly discreteness as well, but they are not best possible and leave us interesting
open questions.
Takumi Noda (Nihon Univ.)
Title: On generalized Lipschitz-type formulae and applications II
Abstract: Let s = σ + it be a complex variable and let H + = {z ∈ C | 0 < arg(z) < π}

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and H − = {z ∈ C | − π < arg(z) < 0} be the complex half planes. The Lipschitz
formula is known as the following equality∞∞X(−2πi)k X k−1−k(z + m) =nexp(2πinz)(z ∈ H + ),(k−1)!
m=−∞
n=1
for positive integer k, and is also known as one of the key ingredients for obtaining theFourier expansion of the Eisenstein series.
The purpose of this talk is to describe a double analogue of the Lipschitz formula via thetwisted Mellin-Barnes integral transformation. We also deﬁne a class of double Eisensteinseries on H + × H − , which includes the real-analytic Eisenstein series for SL(2, Z), andgive the transformation formula by using the generalized Lipschitz-type formula.
Masanori Katsurada (Keio Univ.)
Title: Shintani zeta-functions of several variables and Lauricella hypergeometric functionsAbstract: Let n ≥ 1 be an integer, s and sj (j = 1, . . . , n) complex variables, and aj , λand λj real numbers with aj > 0 (j = 1, 2). The vectorial notation x = (x1 , . . . , xm ) withthe abbreviation hxi = x1 + · · · + xn for any integer m ≥ 1 and any complex numbers
xi (i = 1, . . . , m) is used hereafter. We further write e(λ) = e2πiλ , and let zj be complex
parameters with | arg zj | < π (j = 1, . . . , n). The main object of this talk is the Shintanizeta-function φen of n-variables s = (s1 , . . . , sn ), in the form(1)

φen (s, a, λ; z) =∞Xl1 ,l2 =0nYe(λ1 l1 + λ2 l2 ) {a1 + l1 + (a2 + l2 )zj }−sj ,j=1which converges absolutely for Rehsi > 2. We write 1 = (1, . . . , 1). Then the ‘diagonal’
case s = s1 of (1) is reduced to the one variable (double) zeta-function φn (s, a, λ; z) =φen (s1, a, λ; z), whose (full) multiple form was ﬁrst introduced and studied by T. Shintaniin 1976.
It is the principal aim of this talk to present asymptotic aspects of φen (s, a, λ; z) whenzj (j = 1, . . . , n) are both small and large; this leads us to show complete asymptotic
expansions of φen (s, a, λ; z) in the ascending order of zn as zn → 0, and also in thedescending order of zn as zn → ∞, both through the sectorial region | arg zn − θ0 | < π/2
for any angle θ0 ﬁxed with |θ0 | < π/2, while other zj ’s are in the same sector uponsatisfying the conditions zj ³ zn (j = 1, . . . , n − 1). Here A ³ B means that both A ¿ Band A À B hold. Several applications of our main formulae will further be given.