Instructor: Dr. Árpád TÓTH

Text: Jeff Stopple: A Primer of Analytic Number Theory: From Pythagoras to Riemann, Cambridge University Press, 2003

Prerequisites: General mathematical experience of the undergraduate level is expected. This includes elementary algebra (Abelian groups, vector spaces, systems of linear equations) and calculus (limits, derivatives, integration, infinite series).
A first course in number theory (divisibility, congruences, Chinese Remainder Theorem, primitive roots and power residues) and a course of complex function theory (analytic functions, continuation, power series, complex line integrals, calculation of residues) are useful, although the basic concepts of the theories and theorems applied will always be explained. Taking CLX parallel to this course is enough.

Course description: This is an introductory course on analytic number theory at the undergraduate level. Analytic number theory deals with properties of integers accessible with tools of analysis. The central problem is the distribution of prime numbers among the integers. There is a surprising connection between the primes and the zeros of the Riemann zeta-function. We will follow the history and development of a beautiful discipline, rich in problems, methods and ideas.
After taking this course you will know how analysis is used in other parts of mathematics (as opposed to being developed for its own sake), how calculus is extended to complex numbers and what the connection between the zeros of the Riemann zeta-function and the primes is.

Topics:
Short review. Basic notions and theorems about divisibility, primes and congruences.

Arithmetical functions. Order of magnitude and mean values. Möbius function, elementary prime number estimates.

Series and products, esp. Dirichlet series, Euler products and Perron's type formulas.

Riemann's memoir. The z(s)-function, analytic properties of the zeta-function, explicit formulas connecting primes and the zeros of the Riemann zeta-function.

Extra topics depending on the interest of students (and if time permits) Dirichlet L-function L(s,c), Siegel zeros, Birch-Swinnerton-Dyer conjecture, other prime number sums, the Goldbach--Vinogradov theorem, the Polya--Vinogradov inequality, the large sieve, the Bombieri--Vinogradov theorem.