清华主页 EN
导航菜单

Finite Euler products and the Riemann Hypothesis

来源: 01-10

时间:Tue., 20:00-21:00, Jan 10, 2023

地点:BIMSA 1118 Zoom ID: 293 812 9202 Passcode: BIMSA

组织者:Hansheng Diao, Yueke Hu, Emmanuel Lecouturier, Cezar Lupu

主讲人:Steve M. Gonek (University of Rochester)

Abstract

We investigate approximations of the Riemann zeta function by truncations of its Dirichlet series and Euler product, and then construct a parameterized family of non-analytic approximations to the zeta function. Apart from a few possible exceptions near the real axis, each function in the family satisfies a Riemann Hypothesis. When the parameter is not too large, the functions have roughly the same number of zeros as the zeta function, their zeros are all simple, and they repel. In fact, if the Riemann hypothesis is true, the zeros of these functions converge to those of the zeta function as the parameter increases, and between zeros of the zeta function the functions in the family tend to twice the zeta function. They may therefore be regarded as models of the Riemann zeta function. The structure of the functions explains the simplicity and repulsion of their zeros when the parameter is small. One might therefore hope to gain insight from them into the mechanism responsible for the corresponding properties of the zeros of the zeta function.


Speaker

Prof. Gonek’s research interests are in the field of analytic number theory, particularly multiplicative number theory, the theory of the Riemann zeta-function, L-functions, and the distribution of prime numbers. Some of his work has focused on moments of the Riemann zeta-function, discrete mean value theorems for the zeta-function and L-functions, and the development and application of random matrix models for the zeta-function. One goal of this work is to better understand the behavior (the distribution of zeros, maximal order, and so on) of the zeta and L-functions themselves. Another is to determine connections between these behaviors and various arithmetical problems. Prof. Gonek has also worked on questions relating to the distribution of multiplicative inverses and primitive roots in residue classes modulo a prime.

返回顶部
相关文章
  • A constructive proof of finite time blowup of 3D incompressible Euler equations with smooth data

    Abstract:Whether the 3D incompressible Euler equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this talk, we will present a new exciting result with Dr. Jiajie Chen in which we prove finite time blowup of the 2D Boussinesq and 3D Euler equations with smooth initial data and boundary. There are several essentia...

  • Geodesic Lévy Flight and the Foraging Hypothesis

    Abstract:The Lévy Flight Foraging Hypothesis is a widely accepted dogma which asserts that animals using search strategies allowing for long jumps, also known as Lévy flights, have an evolutionary advantage over those animals using a foraging strategy based on continuous random walks modelled by Brownian motion. However, recent discoveries suggest that this popular belief may not be true in so...