# Number Theory Seminar

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

March 16 at 15:00 - 17:30, 2019

Dr. Yasuo Ohno
(Professor, Mathematical Institute, Tohoku University)

Dr. Yuta Suzuki
(JSPS Research Fellow, Graduate School of Mathematics, Nagoya University)

Venue: Seminar Room #160

Event Official Language: English

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

February 25 at 16:20 - 17:20, 2019

Prof. Hidehiko Mishou (Tokyo Denki University)

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).

Venue: Seminar Room #160

Event Official Language: English

## Symmetric Tornheim double zeta functions

February 25 at 15:00 - 16:00, 2019

Dr. Takashi Nakamura (Tokyo University of Science)

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.

Venue: Seminar Room #160

Event Official Language: English

## Probability density functions attached to zeta functions

December 6 at 16:00 - 17:00, 2018

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.

Venue: Seminar Room #160

Event Official Language: English

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

November 22 at 11:40 - 12:40, 2018

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.

Venue: Seminar Room #160

Event Official Language: English

## Generating functions of CM & RM values

November 22 at 10:30 - 11:30, 2018

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.

Venue: Seminar Room #160

Event Official Language: English

## Relations between fractal dimensions and arithmetic progressions

October 23 at 11:35 - 12:35, 2018

Mr. Kota Saito (Nagoya University)

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.

Venue: Large Meeting Room, 2F Welfare and Conference Building (Cafeteria)

Event Official Language: English

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

October 23 at 10:30 - 11:30, 2018

Mr. Wataru Takeda (Nagoya University)

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).

Venue: Large Meeting Room, 2F Welfare and Conference Building (Cafeteria)

Event Official Language: English

## Number Theory Seminar

July 24 at 10:00 - 12:35, 2018

Dr. Masataka Ono
(Keio University)

Dr. Shingo Sugiyama
(Nihon University)

Mr. Yoshinosuke Hirakawa
(Keio University)

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

Venue: Seminar Room #160

Event Official Language: English