Analytic Methods in Number Theory

Analytic Methods in Number Theory

Analytic Methods in Number Theory

Seminar at FU Berlin, Winter Term 2015-2016

Lei Zhang

Introduction

In this seminar we are going to discuss the applications of analytic methods to number theory. The first aim the seminar is to prove Dirichlet's theorem on arithmetic progressions which states that: given any two positive coprime integers \(a\), \(m\), the set of numbers \(\{a, a+m, a+2m, a+3m,\cdots \}\) contains infinitely many prime numbers. The proof resorts to Dirichlet series, Riemann Zeta functions, Dirichlet \(L\)-function which are very basic tools in the study of analytic number theory. Then we are going to study Modular Forms. Modular forms are holomorphic functions on the upper half plan satisfying certain conditions with respect to some group actions. It is another very important tool to number theory.

Prerequisites

The prerequests for this seminar are rather few. A certain familiarity with undergraduate level real and complex analysis is enough.

Program

You can find the seminar program here.
People who are interested in giving a talk please send an email to me at l.zhang@fu-berlin.de.

DateTitleSpeaker
14/10/2015 Dirichlet's Theorem on Arithmetic Progressions Lei Zhang
21/10/2015 Group Characters (I) Gretar Amazeen
28/10/2015 Group Characters (II) Gretar Amazeen
04/11/2015 Dirichlet Series Yumeng Li
11/11/2015 The Zeta Function Lei Zhang
18/11/2015 The \(L\)-Functions Lei Zhang
25/11/2015 The Dirichlet Theorem Lei Zhang
02/12/2015 Modular Groups Gretar Amazeen
09/12/2015 Modular Functions Lei Zhang
16/12/2015 The Zeros and Poles of a Modular Function Yumeng Li
06/01/2016 The Space of Modular Forms and Modular Invariant Lei Zhang
13/01/2016 Series Expensions Hao Yun
20/01/2016 Hecke Operators (I) Hao Yun
27/01/2016 Hecke Operators (II) Hao Yun
03/02/2016 Theta functions (I) Gretar Amazeen
10/02/2016 Theta functions (II) Gretar Amazeen

Other Information

Place: SR 130/A3 Seminarraum (Hinterhaus) (Arnimallee 3-5)

Date: Wednesday 16:00-18:00

First Appointment: 14.10.2015

Seminar Language: English