アブストラクト - 大東文化大学

解析的整数論とその周辺
Analytic Number Theory and Related Areas
Date :
Place :
Organizers :
November 4 (Wed) – November 6 (Fri), 2015
Room 420, Research Institute for Mathematical Sciences,
Kyoto University, Kyoto, JAPAN
神谷諭一 (大東文化大学)
Yuichi Kamiya (Daito-Bunka University)
石川秀明 (島根大学)
Hideaki Ishikawa (Shimane University)
Abstract of talks
[November 4 (Wed)]
9:40 – 10:20
講演者 : 宗野惠樹 (城西大学)
Speaker : Keiju Sono (Josai University)
タイトル : 素数および概素数のギャップについて
Title : Small gaps between the set of primes and almost primes
Abstract : 相異なる 2 つの素数の積で表される素数を E2 -number という. この講演では, MaynardTao の手法を E2 -number に適用することで得られる素数および E2 -number の分布に関するいく
つかの定理を述べる.
In this talk, we introduce several theorems on the distribution of the set of primes and
E2 -numbers (the set of products of two distinct primes), which are obtained by applying the
multi-dimensional sieve invented by Maynard and Tao.
10:30 – 11:10
講演者 : 金子元 (筑波大学)
Speaker : Hajime Kaneko (University of Tsukuba)
タイトル : 有界な非負整数を係数にもつベキ級数の値の数論的性質
Title : Arithmetical properties of the values of power series
Abstract : Bailey, Borwein, Crandall, Pomerance は代数的無理数の 2 進展開の digit を研究し,
その結果を応用することにより, ベキ級数のある有理点における値の超越性の新しい判定法を
得た. その結果彼らは, 以前の手法では得られることのできない超越数の具体例を与えた.
本講演では, 代数的無理数の 2 進展開に関する彼らの手法を一般化することにより, Pisot 数,
Salem 数を用いたベータ展開の digit を調べる. 我々は, 代数的数のベータ展開に関する研究の
応用として, ベキ級数のある代数点における値の超越性の新しい判定法を得た.
さらに本講演では, 有界な非負整数を係数にもつベキ級数のある代数点における値の代数的
∑∞
独立性および線形独立性の新しい判定法も紹介する. 特に, m=0 β −vm という形をした無限和
1
の数論的性質を考察する. ただし, β は Pisot 数または Salem 数であり, 非負整数からなる数列
vm (m = 0, 1, . . .) は以下の 2 条件を満たすとする:
1. 十分大きなすべての m に対して, vm+1 > vm .
)
(
2. 任意の正の実数 R に対して, limm→∞ vm · m−R = ∞.
Bailey, Borwein, Crandall, and Pomerance investigated the digits in the binary expansions of
algebraic irrational numbers. As applications, they gave new criteria for the transcendence of
the values of power series at certain rational points. Consequently, they obtained new examples
of transcendental numbers.
In this talk, we investigate the β-expansions of algebraic numbers in the case where β is a
Pisot or Salem number, generalizing their methods. As applications, we deduce new criteria for
the transcendence of the values of power series at certain algebraic points.
Moreover, we introduce new criteria for the algebraic independence and linear independence
of the values of power series with bounded nonnegative coeﬃcients. In particular, we consider
∑
−vm
the arithmetical properties of power series of the form ∞
, where β is a Pisot or Salem
m=0 β
number and vm (m = 0, 1, . . .) is a sequence of nonnegative integers such that vm+1 > vm for
any suﬃciently large m and that
vm
lim R = ∞
m→∞ m
for any positive real number R.
11:20 – 12:00
Speaker : Takao Komatsu (Wuhan Univerisy)
Title : Several generalizations and applications of incomplete poly-Bernoulli numbers
and incomplete poly-Cauchy numbers
Abstract : By using the restricted and associated Stirling numbers of the first kind and the
restricted and associated Stirling numbers of the first kind, we define the generalized incomplete
poly-Cauchy numbers and the generalized incomplete poly-Bernoulli numbers, respectively. We
also study their properties, relations, and applications.
13:50 – 14:30
講演者 : 桂田昌紀 (慶應義塾大学)
Speaker : Masanori Katsurada (Keio University)
タイトル : Lerch ゼータ関数に付随した Laplace-Mellin, Riemann-Liouville 変換の漸近展開
Title : Asymptotic expansions for the Laplace-Mellin and Riemann-Liouville transforms
of Lerch zeta-functions
Abstract : The Laplace-Mellin and Riemann-Liouville type transforms are first introduced in a
generic manner under certain natural settings. It is then shown that complete asymptotic expansions exist for these transforms applied to Lerch zeta-functions, when the pivotal parameter
z in the transformations tends to both 0 and ∞ through the sector | arg z| ≤ π/2 including the
critical line Re z = 1/2. Several applications of these results are also to be given.
2
14:40 – 15:20
Speaker : Thomas Oliver (University of Bristol)
Title : Classification of Low Degree L-Data
Abstract : The notion of L-Data was introduced recently by Booker, generalising the axiomatic
approach of the Selberg class. To each L-data, one associates a real number invariant which
is expected to be integral. In this talk we will outline how to classify L-data of degree < 2,
simultaneously generalising the work of Booker and Kaczorowski-Perelli in the case of the
Selberg class.
15:40 – 16:10
講演者 : 大音智弘 (筑波大学)
Speaker : Tomohiro Ooto (University of Tsukuba)
Title : Diophantine exponents for formal power series over a finite field
Abstract : We consider two diophantine exponents wn and wn∗ for formal power series over a
finite field. The functions wn and wn∗ are diophantine exponents related to polynomials and
algebraic numbers, respectively. In this talk, we introduce some results related to the spectrum
of the function wn − wn∗ and that of the function wn at algebraic points.
16:20 – 16:50
講演者 : 鈴木雄太 (名古屋大学)
Speaker : Yuta Suzuki (Nagoya University)
タイトル : 等差数列中の素数２つの和に関する平均値について
Title : A mean value on the sum of two primes in arithmetic progressions
Abstract : いままで Goldbach 予想に関しては, 多くの研究者により様々な変種や近似が考えら
れてきました. 本講演では, その中でも等差数列中の素数２つの和についての平均的な考察をし
ます. 特に, ある種の弱い GRH の下で, この問題の表現関数の平均値の明示公式を導出します.
この明示公式は, 先行する Rüppel (2012) の結果の改善と見れるとともに, 普通の Goldbach 予
想に関する Languasco-Zaccagnini (2012) の結果の一般化とも見れます.
Some variants of the Goldbach problem have been studied by many researchers. In this talk,
we consider such a variant with primes in arithmetic progressions, and study about the mean
value of the representation function of this problem. We show a kind of explicit formula for this
mean value under a weakened variant of GRH. This is an improvement of the result of Rüppel
(2012) and a generalization of the result of Languasco–Zaccagnini (2012).
3
[November 5 (Thu)]
9:10 – 9:50
講演者 : 浜畑芳紀 (岡山理科大学)
Speaker : Yoshinori Hamahata (Okayama University of Science)
Title : The transformations of a series in function fields
Abstract : Let
∞
∏
(
)
πiz/12
η(z) = e
1 − e2πinz
(Im(z) > 0)
n=1
be the Dedekind η-function. R. Dedekind
(
)described the transformation of log η(z) under the
a b
substitution z 0 = (az + b)/(cz + d),
∈ SL2 (Z). To be more exact, he proved that for
c d
(
)
a b
∈ SL2 (Z) with a 6= 0, c > 0,
c d
(
log η
az + b
cz + d
)
1
= log η(z) + log
2
(
cz + d
i
)
+
πi(a + d)
− πiD(a, c),
12c
(1)
where D(a, c) is the Dedekind sum defined by
1 ∑
cot
D(a, c) =
4c k=1
c−1
(
πak
c
)
(
cot
πk
c
)
for coprime integers a and c > 0. We can use (1) to prove the so-called reciprocity law given by
(
)
1
1 a c
1
D(a, c) + D(c, a) = − +
+ +
4 12 c a ac
for coprime positive integers a and c.
An analogy exists between number fields and function fields. For example, A := Fq [T ], K :=
Fq (T ), and K∞ := Fq ((1/T )) are analogous to Z, Q, and R, respectively. A few years ago, we
introduced a function field analog s(a, c) of D(a, c), and established its reciprocity law. In this
talk, we use the Dedekind sum s(a, c) in function fields to(describe
) the transformation of a
a b
certain series under the substitution z 0 = (az + b)/(cz + d),
∈ GL2 (A).
c d
4
10:00 – 10:40
講演者 : 名越弘文 (群馬大学)
Speaker : Hirofumi Nagoshi (Gunma University)
Title : Zeros of the L-function attached to a cusp form and some applications of Selberg’s
orthogonality
Abstract : Let f (z) be a holomorphic cusp form of even integral weight k ≥ 12 for SL2 (Z).
The normalized L-function L(s, f ) attached to f (z) satisfies a certain functional equation whose
critical line is Re s = 1/2. According to the Generalized Riemann Hypothesis, it is believed
that if f (z) is a Hecke eigenform then L(s, f ) has no zeros in the strip 1/2 < Re s < 1. In
contrast, in this talk we will show that if f (z) is not a Hecke eigenform then L(s, f ) has zeros
in any strip σ1 < Re s < σ2 with 1/2 < σ1 < σ2 < 1. Actually a stronger result can be proved.
The proof uses Selberg’s orthogonality among other things.
Two other applications of Selberg’s orthogonality will also be given. We investigate (i) algebraic diﬀerence-diﬀerential independence of L-functions for GLn (AQ ) without assuming the
Generalized Ramanujan Conjecture and (ii) sign changes of Fourier coeﬃcients of a holomorphic cusp form f (z) for SL2 (Z). For example, we obtain unconditionally algebraic diﬀerencediﬀerential independence of all the L-functions attached to irreducible cuspidal automorphic
representations of GLn (AQ ) (1 ≤ n ≤ 4) with unitary central characters.
These three kinds of results (and their proofs) may be considered to indicate that primitive
cusp forms or the associated L-functions are “independent” in the theory of complex analysis,
in the theory of transcendental numbers and functions, and in the theory of Fourier coeﬃcients
of cusp forms, respectively.
(Parts of our results were discussed also at the conference “Diophantine Analysis and Related
Fields 2015” in Japan.)
10:55 – 11:45
Speaker : Régis de la Bretèche (Institut de Mathématiques de Jussieu Paris Rive Gauche)
Title : Friable integers and residues method
Abstract : By a new method resting on a residue computation, we sharpen some of the known
estimates for the counting function of friable integers.
5
13:30 – 14:20
Speaker : Driss Essouabri (Jean Monnet University (Saint-Etienne))
Title : Zeta functions and solutions of Falconer type problems for self-similar subsets of Z n
Abstract : In this talk, I will report on my joint work with Ben Lichtin on Zeta functions of selfsimilar subsets of Z n and their applications. In this work we use the method of zeta functions to
solve Falconer type problems about sets of k-simplices whose endpoints belong simultaneously
to a self-similar subset F of Z n (k ≤ n). More precisely, we study four Euclidean metric
invariants of the simplices, the most basic (and frequently studied) of which is the distance
between endpoints of a 1-simplex. For each, we introduce a zeta function, derive its functional
properties, and apply such information to derive a lower bound on the number of k−simplices
whose endpoints belong simultaneously to a self-similar subset F of Z n and a disc B(x) of a
large radius x.
14:30 – 15:10
講演者 : 藤田育嗣 (日本大学)
Speaker : Yasutsugu Fujita (Nihon University)
タイトル : ディオファントスの３組の４組への拡張可能性
Title : Extendabilities of a Diophantine triple to quadruples
Abstract : A set {a1 , . . . , am } of m positive integers is called a Diophantine m-tuple if ai aj + 1 is
a perfect square for all i, j with i < j. Any Diophantine triple {a, b, c} with a < b < c is known
to be extended to a Diophantine quadruple {a, b, c, d+ }, where d+ = a + b + c + 2abc + 2rst and
√
√
√
r = ab + 1, s = ac + 1, t = bc + 1. Such a quadruple {a, b, c, d+ } is called regular, and
any Diophantine quadruple is conjectured to be regular, which immediately implies a folklore
conjecture asserting that there exists no Diophantine quintuple.
In the present talk, we show that for a fixed Diophantine triple {a, b, c} with a < b < c, the
number of d such that {a, b, c, d} forms a Diophantine quadruple with d > c is at most 11. The
proof of this theorem needs:
• Rickert’s theorem on simultaneous rational approximation of quadratic irrationals;
• A gap principle due to Okazaki on solutions of simultaneous Pellian equations;
• Baker’s method on linear forms in two and three logarithms.
Relevant results to the theorem will be also presented.
15:30 – 16:10
Speaker : Tim Trudgian (The Australian National University)
Title : Bounding Diophantine Quintuples
Abstract : The set {1, 3, 8, 120} has the property that the product of any two of its elements is
one less than a perfect square. A five-element set with this property is said to be a Diophantine
quintuple: it is conjectured that there are no such sets. It is known that there are only finitely
many Diophantine quintuples. I shall highlight recent advances in this area.
6
16:20 – 16:50
講演者 : 一階智弘 (名古屋大学)
Speaker : Tomohiro Ikkai (Nagoya University)
タイトル : 素数を法とした Pascal の三角形の直線による裁断
Title : Cutting Pascal’s triangle modulo a prime by a straight line
Abstract : 二項係数が並んだ Pascal の三角形において，与えられた素数で割り切れないような
ものだけに注目すると，自己相似性を持った集合が現れる．Essouabri はこの集合 (素数を法と
して見た Pascal の三角形) に関する，二変数多項式を用いたある種の Dirichlet 級数が定める有
理型関数を研究した．この関数はフラクタル幾何学の問題と関連し，絶対収束軸上に実でない
極を持つかどうかということが一つの問題となっている．Essouabri は多項式が最も単純な形を
しているときに，このような極を持つことを証明した．証明において鍵となるのは，素数を法
として見た Pascal の三角形の点の個数に関する，Stein や Wilson の評価である．今回の発表で
は，素数を法として見た Pascal の三角形の点の数え方を変えることにより，Essouabri の結果
が他の数例に拡張できることを紹介する．
Picking binomial coeﬃcients which cannot be divided by a given prime from Pascal’s triangle,
we find that they form a set with self-similarity. Essouabri studied the meromorphic functions
defined by a kind of Dirichlet series, associated to the above set (Pascal’s triangle modulo a
prime) with two-variable polynomials. Such functions are related to a problem in fractal geometry, and it is a problem whether they have a non-real pole on their axes of absolute convergence.
Essouabri proved that there exists a non-real pole on its axis of absolute convergence in the
case when polynomials are the simplest. Keys of the proof are Stein’s and Wilson’s estimates
with respect to the number of points in Pascal’s triangle modulo a prime. In this talk, I explain
that Essouabri’s result can be extended to several cases by counting points of Pascal’s triangle
modulo a prime in other way.
7
[November 6 (Fri)]
9:10 – 9:50
講演者 : 立谷洋平 (弘前大学)
Speaker : Yohei Tachiya (Hirosaki University)
タイトル : ある種の無限級数の線形独立性について
Title : Linear independence results for various infinite series
Abstract : In this talk we present some recent linear independence results for certain Lambert
series and for certain reciprocal sums of Fibonacci numbers. Our method is based on the original
approach of Erdős together with the results on primes in an arithmetic progression by Alford,
Granville, and Pomerance.
This is a joint work with Hiromi Ei (Hirosaki Univ.) and Florian Luca (Univ. of Witwatersrand).
10:00 – 10:50
Speaker : Stéphane Louboutin (University Aix-Marseille)
Title : Real zeros of Dedekind zeta functions
Abstract : Building on Stechkin and Kadiri’s ideas we derive an explicit zero-free region of the
real axis for Dedekind zeta functions of number fields. We then explain how this new region
enables us to improve upon the previously known explicit lower bounds for class numbers of
number fields and relative class numbers of CM-fields.
11:00 – 11:50
Speaker : Gautami Bhowmik (Université de Lille 1)
Title : Non-vanishing of L-functions
Abstract : We will present results old and new results on the non vanishing of the symmetric
square L-function.
13:30 – 14:00
講演者 : 岡本卓也 (日本工業大学)
Speaker : Takuya Okamoto (Nippon Institute of Technology)
タイトル : Dirichlet の L 関数の平均値
Title : On the mean value of the Dirichlet L-functions
Abstract : The mean values of Dirichlet L-functions has been studied extensively by a lot of
mathematicians. In this talk, we consider the generalized mean values of Dirichlet L-functions,
and give the explicit formula for the generalized mean values of Dirichlet L-functions which
are expressed in terms of the Riemann zeta-function, the Euler function and Jordan’s totient
functions.
8
14:10 – 14:50
Speaker : Christoph Aistleitner (University Linz)
Title : Extreme values of the Riemann zeta function via the resonance method
Abstract : The problem of estimating the order of the maximal modulus of the Riemann
zeta function along vertical lines in the complex plane is a classical and very diﬃcult problem
in analytic number theory. Of particular interest are the behaviour of the zeta function on
the critical line, and along lines having real part between 1/2 and 1. The best known lower
bounds are due to Montgomery (1977), who used Diophantine approximation and zero-density
estimates. Recently Soundararajan (2008) introduced a new method, called the ”resonance
method”, which works in a completely diﬀerent way. In this talk we explain the connection
between the resonance method and certain sums of greatest common divisors (GCD sums),
which play a important role in metric number theory. We show how these GCD sums can be
used to construct a ”long” resonator function, and which diﬃculties have to be addressed.
15:00 – 15:30
講演者 : 門田慎也 (名古屋大学)
Speaker : Shinya Kadota (Nagoya University)
タイトル : 多重ゼータ値に対するパラメータを持つ重みつき和公式
Title : A weighted sum formula for multiple zeta values with some parameters
Abstract : 多重ゼータ値とは自然数の組に対してある多重級数によって定められる実数値であ
り、現在では、それらの間の関係式がたくさん知られている。今回の講演では、パラメータを
重みに持つような重みつき和公式についてお話しする。
Multiple zeta values are the real numbers which are defined by certain multiple series for the
tuple of natural numbers and many relations among them are known currently. In this talk, I
would like to talk on a weighted sum formula with some parameters as weights.
15:40 – 16:10
講演者 : 池田創一 (芝浦工業大学)
Speaker : Soichi Ikeda (Shibaura Institute of Technology)
Title : Double analogue of Hamburger’s theorem
Abstract : ハンバーガーの定理はリーマンゼータ関数をその関数等式で特徴づける古典的結果
である。最近、松本はオイラーの二重ゼータ関数の関数等式を発見した。しかし、オイラーの
二重ゼータ関数を関数等式で特徴づける研究はなされてこなかった。本講演ではオイラーの二
重ゼータ関数の関数等式による特徴づけについて紹介する。
Hamburger’s theorem is a classical result which gives a characterization of the Riemann zetafunction by functional equation. Recently, Matsumoto found a functional equation for the Euler
double zeta-function. However, characterization of the Euler double-zeta function by functional
equations has not been studied. In this talk, we discuss a characterization of the Euler double
zeta-function by functional equations.
9

Download Report