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Number Theory Seminar

9 イベント

セミナー Number Theory Seminar
On special values of the multiple zeta functions of Arakawa-Kaneko type

2019年3月16日(土) 15:00 - 17:30

大野泰生 (東北大学理学部数学科教授)
鈴木雄太 (名古屋大学大学院多元数理科学研究科学振特別研究員)

15:00~16:00: Yasuo Ohno (Tohoku University) "On special values of the multiple zeta functions of Arakawa-Kaneko type" Arakawa-Kaneko'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.

会場: セミナー室 (160号室)

イベント公式言語: 英語
セミナー Number Theory Seminar
Joint value distribution of quadratic L-functions (joint work with Hirofumi Nagoshi)

2019年2月25日(月) 16:20 - 17:20

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
会場: セミナー室 (160号室)

イベント公式言語: 英語
セミナー Number Theory Seminar
Symmetric Tornheim double zeta functions

2019年2月25日(月) 15:00 - 16:00

中村隆 (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.

会場: セミナー室 (160号室)

イベント公式言語: 英語
セミナー Number Theory Seminar
Probability density functions attached to zeta functions

2018年12月6日(木) 16:00 - 17:00

Masahiro Mine (東京工業大学)

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.

会場: セミナー室 (160号室)

イベント公式言語: 英語
セミナー Number Theory Seminar
On A_2-liftings of sum formulas and Bowman-Bradley type formulas for finite multiple zeta values

2018年11月22日(木) 11:40 - 12:40

関真一朗 (東北大学)

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.

会場: セミナー室 (160号室)

イベント公式言語: 英語
セミナー Number Theory Seminar
Generating functions of CM & RM values

2018年11月22日(木) 10:30 - 11:30

松坂俊輝 (九州大学)

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.

会場: セミナー室 (160号室)

イベント公式言語: 英語
セミナー Number Theory Seminar
Relations between fractal dimensions and arithmetic progressions

2018年10月23日(火) 11:35 - 12:35

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.

会場: 統合支援施設（第一食堂） 2階大会議室

イベント公式言語: 英語
セミナー Number Theory Seminar
Generalized Erdös and Obláth theorem for polynomial-factorial Diophantine equations

2018年10月23日(火) 10:30 - 11:30

武田渉 (名古屋大学)

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).

会場: 統合支援施設（第一食堂） 2階大会議室

イベント公式言語: 英語
セミナー Number Theory Seminar
Number Theory Seminar: 3 Talks

2018年7月24日(火) 10:00 - 12:35

小野雅隆 (慶應義塾大学)
杉山真吾 (日本大学)
平川義之輔 (慶應義塾大学)

This seminar is aimed at scientists in general, not only to mathematicians. 10:00-10:45 Title: Multiple zeta functions associated with 2-colored 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 2-colored 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 2-colored rooted tree and discuss its analytic properties, for example, the possible singularities and functional equations. 10:55-11:40 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 L-functions, Hurwitz class numbers. We will start fundamental notions on modular forms and trace formulas of Hecke operators for non-experts, and introduce our results on a generalization of Serre’s vertical Sato-Tate law. Some results in this talk are based on a joint work with Masao Tsuzuki (Sophia University). 11:50-12:35 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 "combinatorial-analytic" 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.

会場: セミナー室 (160号室)

イベント公式言語: 英語

9 イベント