Number Theory Seminar
9 events

Seminar
On special values of the multiple zeta functions of ArakawaKaneko type
March 16 (Sat) at 15:00  17:30, 2019
Yasuo Ohno (Professor, Mathematical Institute, Tohoku University)
Yuta Suzuki (JSPS Research Fellow, Graduate School of Mathematics, Nagoya University)15:00~16:00: Yasuo Ohno (Tohoku University) "On special values of the multiple zeta functions of ArakawaKaneko type" ArakawaKaneko's zeta function is a kind of generalization of the Riemann zeta function by using polylogarithms. I am planning to introduce two topics on combinatorics around its values at positive or negative integral points. This talk is based on joint work with N. Kawasaki. 16:20~17:20: Yuta Suzuki (Nagoya University) "On relatively prime amicable pairs" A famous Greek mathematician, Pythagoras tried to find symbolical meanings in numbers. One famous example of such symbolism in numbers is amicable pair, which was introduced as a symbol of friendship. A pair of positive integers $(M,N)$ is called an amicable pair if the sum of all divisors of $M$ except $M$ itself is equal to $N$ and the sum of all divisors of $M$ except $N$ itself is also equal to $M$. The smallest example is $(220,284)$. Even amicable numbers are introduced more than 20 centuries ago, most of their properties are still unknown. For example, the infinitude of amicable pairs has not yet been proven. In this talk, we pick up a conjecture of Gmelin (1917), which claims that there is no relatively prime amicable pairs, and try to introduce some atmosphere of "modern elementary number theory". In particular, we improve Pollack's partial result (2015) on Gmelin's conjecture.
Venue: Seminar Room #160
Event Official Language: English

Seminar
Joint value distribution of quadratic Lfunctions (joint work with Hirofumi Nagoshi)
February 25 (Mon) at 16:20  17:20, 2019
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
Venue: Seminar Room #160
Event Official Language: English

Seminar
Symmetric Tornheim double zeta functions
February 25 (Mon) at 15:00  16:00, 2019
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

Seminar
Probability density functions attached to zeta functions
December 6 (Thu) at 16:00  17:00, 2018
Masahiro Mine (Tokyo Institute of Technology)
The study of the valuedistribution 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 valuedistributions of zeta functions. The density functions, which were named ``Mfunctions'' 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

Seminar
On A_2liftings of sum formulas and BowmanBradley type formulas for finite multiple zeta values
November 22 (Thu) at 11:40  12:40, 2018
Shinichiro Seki (Tohoku University)
Both the sum formula and BowmanBradley's theorem for multiple zeta values are well known. Recently, Saito and Wakabayashi proved counterparts of these two formulas for Afinite multiple zeta values. In this talk, I will explain that A_2liftings of some parts of SaitoWakabayashi's results have simple forms using SekiBernoulli 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

Seminar
Generating functions of CM & RM values
November 22 (Thu) at 10:30  11:30, 2018
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, DukeImamogluToth 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 LagariasRhoades on the Kronecker limit formula, and give a generalization of DukeImamogluToth’s work for any polyharmonic weak Maass form.
Venue: Seminar Room #160
Event Official Language: English

Seminar
Relations between fractal dimensions and arithmetic progressions
October 23 (Tue) at 11:35  12:35, 2018
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 singleauthor work.
Venue: Large Meeting Room, 2F Welfare and Conference Building (Cafeteria)
Event Official Language: English

Seminar
Generalized Erdös and Obláth theorem for polynomialfactorial Diophantine equations
October 23 (Tue) at 10:30  11:30, 2018
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

Seminar
Number Theory Seminar: 3 Talks
July 24 (Tue) at 10:00  12:35, 2018
Masataka Ono (Keio University)
Shingo Sugiyama (Nihon University)
Yoshinosuke Hirakawa (Keio University)This seminar is aimed at scientists in general, not only to mathematicians. 10:0010:45 Title: Multiple zeta functions associated with 2colored rooted trees Speaker: Dr. Masataka Ono (Keio University) Abstract: In our recent work, we introduced a combinatorial object and finite sum associated with them which we call finite multiple zeta values associated with 2colored rooted trees and gave a unified interpretation to some types of finite multiple zeta values. In this talk, we introduce multiple zeta function associated with 2colored rooted tree and discuss its analytic properties, for example, the possible singularities and functional equations. 10:5511:40 Title: Modular forms and trace formulas with applications to equidistributions of their Fourier coefficients Speaker: Dr. Shingo Sugiyama (Nihon University) Abstract: Modular forms are interesting objects in number theory as they are related to arithmetic problems. Trace formulas of Hecke operators acting on modular forms are very useful tools to study arithmetic invariants: Fourier coefficients, special values of Lfunctions, Hurwitz class numbers. We will start fundamental notions on modular forms and trace formulas of Hecke operators for nonexperts, and introduce our results on a generalization of Serre’s vertical SatoTate law. Some results in this talk are based on a joint work with Masao Tsuzuki (Sophia University). 11:5012:35 Title: On a generalization of Dobinski's formula Speaker: Yoshinosuke Hirakawa (Keio University) Abstract: Dobinski's formula is a very classical formula, which expresses the Bell number as an infinite series. Here, the Bell number is the number of partitions of a finite set. Such a "combinatorialanalytic" formula should lead us to more beautiful number theory. In this talk, we introduce a generalization of Dobinski's formula by means of a certain multiple generalization of the exponential function.
Venue: Seminar Room #160
Event Official Language: English
9 events