Description

Analytic number theory is arguably one of the oldest branches of mathematics. The earliest examples go as far back as the discovery of Pythagorean triples by Babylonians. One of the central topics in analytic number theory is the theory of Riemann zeta function and its subsequent generalisations, called as the L-functions. ''The Riemann Hypothesis'', arguably one of the most famous open problems in number theory, has driven much of the research in the field. In the recent years, connections of the L-functions has been established from the links with distribution of primes, random matrix theory, representation theory to mathematical physics.

This project will focus on learning some essential topics in modern analytic number theory, by focusing on the the Riemann zeta function, Dirichlet series and L-functions. Roughly, we could look at the following topics:

• Riemann zeta function and analytic continuation
  
• Dirichlet series and theory of general L-functions
  
• Approximate functional equation
  
• Prime number theorem
  
• Algorithms for computing the zeta function
• Understanding the Riemann hypothesis, Lindelof hypothesis and ''subconvex bounds''
  

Prerequisites

Resources