Title & Abstract
2023 / 08 / 21 (Aug. 21st)
[Speaker] Yoonbok Lee
[Title] Selberg's central limit theorem of L-functions and the second moment of log L.
[abstract] Selberg's central limit theorem says that the logarithm of the Riemann zeta function has a Gaussian distribution in the complex plane on and near the critical line. We find an asymptotic expansion of a multi-dimensional version of Selberg's central limit theorem for L-functions near the critical line. To prove it for sigma very near 1/2, we estimate the second moment of log L.
[Speaker] Shota Inoue
[Title] Large deviations estimates of the Riemann zeta-function on the critical line
[abstract] Selberg established the central limit theorem for the Riemann zeta-function on the critical line. His limit theorem was developed by several mathematicians. For example, Radziwi l l gave a large deviation estimate for the limit theorem. In his work, he also suggested a conjecture for large deviations, including Keating-Snaith Conjecture for moments of the Riemann zeta-function. The speaker improved Radziwi l l’s estimate and resolve his conjecture partially. In this talk, the speaker will present the result and the key idea of the proof.
[Speaker] Hwajong Yoo
[Title] Modular units on X_0(N)
[abstract] We discuss meromorphic functions on the modular curve X_0(N) which are holomorphic and non-vanishing on the upper half plane. Such functions are well-studied by the work of Kubert and Lang when the modular curves are X(N) or X_1(N). As an application of our study, we partially prove the conjecture that asserts the rational cuspidal subgroup of J_0(N) is equal to the rational cuspidal divisor class group of X_0(N). This talk is based on our joint work with J.-W. Guo, Y. Yang, and M. Yu.
[Speaker] Kenta Endo
[Title] Universality theorem for the iterated integrals of the logarithm of the Riemann zeta-function
[abstract] The universality phenomenon for the Riemann zeta-function was discovered by Voronin in 1975. Since then, the universality theorem has been generalized for a wide class of zeta-functions. In this talk, we present the universality theorem for the iterated integrals of the logarithm of the Riemann zeta-function, which represents a further generalization in a different direction.
[Speaker] Jungyun Lee
[Title] The mean value of the class numbers of cubic function fields
[abstract] We compute the mean value of |L(s,chi)|^2 evaluated at s=1 when chi goes through the primitive cubic Dirichlet characters of A:=F_q[T], where F_q is a finite field with q elements and q \equiv 1 \pmod 3. Furthermore, we find the mean value of the class numbers for the cubic function fields K_m=k(\sqrt[3]{m}), where k:= F_q(T) is the rational function field and m in A is a cube-free polynomial.
(This is a joint work with Yoonjin Lee and Jinjoo Yoo.)
2023 / 08 / 22 (Aug. 22nd)
[Speaker] Aiki Kimura
[Title] On deducing the duality relations from the double shuffle relations among multiple zeta values
[abstract] Multiple zeta values (MZVs) are the values of the Euler-Zagier multiple zeta functions at positive integers and all of them are believed to be irrational numbers. Zagier gave a surprising conjecture about the structure of the algebra generated by all multiple zeta values over the rationals, and he pointed out that the structure is deeply related to important objects in other fields, such as modular forms and knot-invariants. The family of the double shuffle relations is expected to generate all linear relations among multiple zeta values and to be a cue for the conjecture. However, it remains unclear as to whether all duality relations, basic linear relations for MZVs, can be deduced from the double shuffle relations. In this talk, I will present that a certain family of duality relations is deduced from the double shuffle relations. The family is expected to reveal that the double shuffle relations can generate all duality relations.
[Speaker] Shin-ichiro Seki
[Title] Constellation theorems for norm forms
[abstract] We consider an existence problem of constellations on the representation of primes by a d-variable map F : Z d → Z. The case F(x) = x is the Green-Tao theorem, and Tao proved the case F(x, y) = x 2 + y 2 . In this talk, the speaker presents the result for the case where F is a general norm form. This work is based on joint work with Kai, Mimura, Munemasa, and Yoshino.
2023 / 08 / 23rd (Aug. 23rd)
[Speaker] Subong Lim
[Title] Density of modular forms with transcendental zeros
[abstract] Rankin and Swinnerton-Dyer proved that all zeros of the Eisenstein series on the full modular group in the fundamental domain lie in the unit circle. Since then, many studies have been conducted on the zeros of modular forms, especially on the transcendence of zeros of modular forms. For example, Gun, Murty, and Rath showed that if f is a nonzero modular form with algebraic Fourier coefficients, then any zero of f is either CM or transcendental. In this talk, we present the result for the density of modular forms with only transcendental zeros on the complex upper half plane except elliptic points. This is joint work with Dohoon Choi and Youngmin Lee.
[Speaker] Masahiro Mine
[Title] Zeros of the Hurwitz zeta-function with algebraic irrational parameter
[abstract] The Hurwitz zeta-function is defined by a generalized Dirichlet series with a parameter. It is classically known that the Hurwitz zeta-function has infinitely many zeros whose real part is greater than 1, while the Riemann zeta-function has no such zeros by the Euler product. Furthermore, if the parameter is rational or transcendental, the Hurwitz zeta-function has infinitely many zeros whose real part lies between 1/2 and 1. In this talk, we extend the result to many of the Hurwitz zeta-functions with algebraic irrational parameters. This result was derived by resolving a weak version of the Gonek Conjecture on the universality of Hurwitz zeta-function with algebraic irrational parameter.
2023 / 08 / 24th (Aug. 24th)
[Speaker] Yuichiro Toma
[Title] Bounds for the double L-function
[abstract] The double zeta-function and double L-functions are generalizations of the classical Riemann zetafunction and L-functions to functions of two variables, respectively. Analytic properties of the double zeta-function have been studied for the last two decades. In this talk, we report results on upper bound estimates for the double L-function attached to two primitive Dirichlet characters.
[Speaker] Hirotaka Kobayashi
[Title] Mean-square values of the Riemann zeta function over arithmetic progressions
[abstract] The mean values of the Riemann zeta function have been studied for many years. As for the continuous mean values, the big progress has been made. On the other hand, our knowledge on the discrete mean values are limited. Recently, Li and Radziwi l l revealed that the twisted second moments of the Riemann zeta function over arithmetic progressions shows a notable correspondence with the analogous continuous moment. In this talk, we will give the result on the second moments over arithmetic progressions without twist.
[Speaker] Gilyoung Cheong
[Title] The cokernel of a polynomial of a random integral matrix
[abstract] Given a prime p, let P(t) be a non-constant monic polynomial in t over the ring Zp of p-adic integers. Let Xn be the n × n uniformly random (0, 1)-matrix over Zp. We compute the asymptotic distribution of the cokernel of P(Xn) as n goes to infinity. In fact, we shall consider the same problem with a more general random matrix Xn, which also includes the example of a Haar-random matrix. Our work crucially uses a recent work of W. Sawin and M. M. Wood which shows that the moments of finite size modules over any finite ring determine their distribution. This is joint work with Myungjun Yu.
[Speaker] Jungin Lee
[Title] Mixed moments and the joint distribution of random p-adic matrices
[abstract] The moment problem is to determine whether a probability distribution is uniquely determined by its moments. Recently, the moment problem for random groups has been applied to the distribution of random groups, in particular the cokernels of random p-adic matrices. In this talk, we introduce the (mixed) moments of random groups and provide several universality results for the (joint) distribution of the cokernels of random p-adic matrices.