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.