This module is designed to provide an introduction to topics in Analytic Number Theory, including the worst and average case behaviour of arithmetic functions, properties of the Riemann zeta function, and the distribution of prime numbers.

(LO1) Be able to apply analytic techniques to arithmetic functions.

(LO2) Understand basic analytic properties of the Riemann zeta function.

(LO3) Understand Dirichlet characters and L-series.

(LO4) Understand the connection between Ingham's theorem and the Prime Number Theorem.

(S1) Adaptability

(S2) Problem solving skills

(S3) Numeracy

Preliminary review of arithmetic prerequisites, including congruences, arithmetic functions and multiplicative functions. Worst and average case behaviour of the divisor function, mobius and tau functions. Euler summation, harmonic series and counting lattice points inside convex neighbourhoods. The Riemann zeta function and its analytic properties, the residue theorem, entire functions of order one and infinite products, and analytic continuation. Dirichlet characters, L-series and their analytic properties. Dirichlet's Theorem on Primes in Arithmetic Progression. Ingham's Tauberian Theorem. Equivalent formulations of the Prime Number Theorem. Derivation of the Prime Number Theorem from Ingham's Theorem using D.J. Neumann's method.