# Number Theory Seminar

セミナー

## On special values of the multiple zeta functions of Arakawa-Kaneko type

2019年3月16日15:00 - 17:30

イベント公式言語: 英語

セミナー

## Joint value distribution of quadratic L-functions (joint work with Hirofumi Nagoshi)

2019年2月25日16:20 - 17:20

Prof. Hidehiko Mishou (東京電機大学)

In 1975, Voronin established the universality theorem for the Riemann zeta function. Roughly speaking this theorem asserts that any holomorphic function on 1/2<Re(s)<1 can be uniformly approximated by a suitable vertical translation of the Riemann zeta function. In this talk, we state that the joint universality theorem for a set of Dirichlet L-functions associated with real primitive characters holds as we move the modulus of characters. As a corollary of this result, we also establish a joint denseness result for a set of class numbers of imaginary quadratic fields. This is a joint work with Hirofumi Nagoshi (Gunma University).

イベント公式言語: 英語

セミナー

## Symmetric Tornheim double zeta functions

2019年2月25日15:00 - 16:00

Dr. Takashi Nakamura (東京理科大学)

Let $s,t,u \in {\mathbb{C}}$ and $T(s,t,u)$ be the Tornheim double zeta function. We investigate some properties of symmetric Tornheim double zeta functions. As a corollary, we give explicit evaluation formulas for $T(s,t,u)$ in terms of series of the gamma function and Riemann zeta function.

イベント公式言語: 英語

セミナー

## Probability density functions attached to zeta functions

2018年12月6日16:00 - 17:00

Dr. Masahiro Mine (Tokyo Institute of Technology)

The study of the value-distribution of the Riemann zeta function is a classical topic in analytic number theory. In 1930s, Bohr and Jessen proved the existence of a certain limit value regarded as the probability that values of the Riemann zeta function belong to a given region in the complex plane. After Bohr and Jessen, similar results were proved for many other zeta functions. In this talk, I'll talk about density functions of such probabilities attached to the value-distributions of zeta functions. The density functions, which were named M-functions'' by Ihara, are connected with mean values of zeta functions, distributions of zeros of zeta functions, and so on.

イベント公式言語: 英語

セミナー

## On A_2-liftings of sum formulas and Bowman-Bradley type formulas for finite multiple zeta values

2018年11月22日11:40 - 12:40

Dr. Shin-ichiro Seki (Tohoku University)

Both the sum formula and Bowman-Bradley's theorem for multiple zeta values are well known. Recently, Saito and Wakabayashi proved counterparts of these two formulas for A-finite multiple zeta values. In this talk, I will explain that A_2-liftings of some parts of Saito-Wakabayashi's results have simple forms using Seki-Bernoulli numbers. The first part of this talk is a joint work with Shuji Yamamoto. The second part is a joint work with Hideki Murahara and Tomokazu Onozuka.

イベント公式言語: 英語

セミナー

## Generating functions of CM & RM values

2018年11月22日10:30 - 11:30

Dr. Toshiki Matsusaka (Kyushu University)

The special values of the elliptic modular j function j(z) at imaginary quadratic points are known as singular moduli (CM values), and play important roles in algebraic number theory. As a real quadratic analogue, Kaneko (2009) defined the values’ of j(z) at real quadratic points (RM values). In 2011, Duke-Imamoglu-Toth showed that the generating function of the traces of these CM & RM values becomes a harmonic Maass form of weight 1/2. In this talk, I shall introduce a new class called polyharmonic weak Maass forms, inspired by works of Lagarias-Rhoades on the Kronecker limit formula, and give a generalization of Duke-Imamoglu-Toth’s work for any polyharmonic weak Maass form.

イベント公式言語: 英語

セミナー

## Relations between fractal dimensions and arithmetic progressions

2018年10月23日11:35 - 12:35

Mr. Kota Saito (名古屋大学)

In this talk we give estimates for the dimensions of sets in real numbers which uniformly avoid finite arithmetic progressions. More precisely, we say that $F$ uniformly avoids arithmetic progressions of length $k\geq 3$ if there is an $\epsilon>0$ such that one cannot find an arithmetic progression of length $k$ and gap length $\Delta>0$ inside the $\epsilon\Delta$ neighbourhood of $F$. Our main result is an explicit upper bound for the Assouad (and thus Hausdorff) dimension of such sets in terms of $k$ and $\epsilon$. In the other direction, we give examples of sets which uniformly avoid arithmetic progressions of a given length. We also consider higher dimensional analogues of these problems, where arithmetic progressions are replaced with arithmetic patches lying in a hyperplane. As a consequence, we obtain a discretised version of a reverse Kakeya problem': we show that if the dimension of a set in $\mathbb{R}^d$ is sufficiently large, then it closely approximates arithmetic progressions in every direction. The above is a joint work with Fraser and Yu. Finally we show that the converse of `reverse Kakeya problem' does not hold. This is a single-author work.

イベント公式言語: 英語

セミナー

## Generalized Erdös and Obláth theorem for polynomial-factorial Diophantine equations

2018年10月23日10:30 - 11:30

Diophantine equations are equations where only integer solutions are accepted. There are many types of Diophantine equations and many results are known. Our Diophantine equation is of the form x^n+y^n=m!. Erdös and Obláth showed that the Diophantine equation x^2+y^2=m! has only two positive integer solutions (x,y,m)=(1,1,2),(12,24,6). In this talk, the factorial function m! is replaced with a generalized factorial function Π(m) over number fields. Then whether there are infinitely many solutions or not depends on the number field. We give necessary and sufficient condition for existence of infinitely many solutions of x^2+y^2=Π(m). More generally, we introduce an observation for higher degree equation x^n+y^n=Π(m).

イベント公式言語: 英語

セミナー

## Number Theory Seminar

2018年7月24日10:00 - 12:35

This seminar is aimed at scientists in general, not only to mathematicians. 10:00-10:45 "Multiple zeta functions associated with 2-colored rooted trees" Masataka Ono 10:55-11:40 "Modular forms and trace formulas with applications to equidistributions of their Fourier coefficients" Shingo Sugiyama 11:50-12:35 "On a generalization of Dobinski's formula" Yoshinosuke Hirakawa

イベント公式言語: 英語