# RMA

## Seminar

### Prime number theory & the Riemann zeta-function I

Seminar Room 1, Newton Institute

#### Abstract

Lecture 1:-

Unique Factorization Theorem Infinitude of primes Statement of PNT Cramer model Failure of Cramer model

Lecture 2:-

Open questions on primes Recent achievements of prime number theory The Riemann Zeta-function Euler product Analytic continuation and functional equation (via theta function)

Lecture 3:-

Analytic continuation and functional equation (continued) Hadamard product and its logarithmic derivative N(T) and S(T)

Lecture 4:-

N(T) and S(T) (continued) Non-vanishing on the 1-line Proof of PNT

Lecture 5:-

Proof of PNT (continued) Weil type Explicit formulae

Lecture 6:-

Characters Dirichlet L-functions

Pre-requisites:-

Undergrad complex analysis Prpoerties of the gamma function Undergrad algebra (Z is a UFD)

Recommended text:-

Davenport, Multiplicative Number Theory