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.

Date | Title | Speaker |

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