## CS 746: Riemann Hypothesis and Its Applications

### Prerequisites

MTH102, MTH103, or equivalent

### Course Contents:

Riemann Hypothesis is one of the most important unresolved conjectures in mathematics. It connects the distribution of prime numbers with zeroes of Zeta function, defined on the complex plane. A number of algorithms in algebra and numbertheory rely on the correctness of Riemann Hypothesis or its generalizations. This course willdescribe the connection between prime distributions and Zeta function leading to the Riemann Hypothesis, prove Prime Number Theorem along the way, and then describe the generalizations of Riemann Hypothesis and their applications to computer science problems.

#### Topics:

- Prime counting and other arithmetic functions

- Brief overview of complex analysis

- Zeta function definition and basic properties

- Riemann Hypothesis and its relationship with prime counting

- Prime Number Theorem

- Dirichlet L-functions and Generalized Riemann Hypothesis

- Applications of Riemann and Generalized Riemann Hypothesis

- Further extensions of Riemann Hypothesis and its proof for function fields, and elliptic curves over finite fields

### Books and References:

- M. Ram Murty. Problems in Analytical Number Theory. Graduate Text in Mathematics 206, Springer 2001.

- Peter Borwein, Stephen Choi, Brendan Rooney, AnderaWierathmueller. The Riemann Hypothesis. Springer 2006. (Available here).

- Research papers.