Topics in Number Theory    NUM2

Instructor: Dr. Antal BALOG

Text: Classnotes by the lecturer based on selected chapters of Harold Davenport, Multiplicative Number Theory.

Prerequisite: 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 of 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 essential, although the basic concepts of the applied theories and theorems will always be explained. Taking CLX parallel to this course is enough.

Course description: Our aim is to provide a classical introduction to analytic number theory, focused on the connection between the zeros of the Riemann z(s)-function and prime numbers. We will follow the history and development of a beautiful discipline, rich in problems, methods and ideas. The highlights are a proof of Dirichlet's Theorem about the distribution of primes in arithmetic progressions, and of the Prime Number Theorem, but the course is much more than just the proof of two famous theorems.


Short review. Basic notions and theorems about divisibility, primes and congruences, the Legendre symbol and the quadratic reciprocity, primitive roots and discrete logarithm.

Arithmetical functions. Additive and multiplicative functions, convolution, the Möbius  inversion formulas, partial summation, mean value of some functions, elementary prime number estimates.

Dirichlet series. The general theory of Dirichlet series, convergence, Euler products, Perron's type formulas.

Dirichlet's theorem. The prime modulus case, characters, L--functions, the general case, primitive characters and Gaussian sums. (If time permits: quadratic forms, Dirichlet's class number formula.)

Riemann's memoir.  The z(s)-function, analytic properties of z(s) and L(s,c), integral functions of order 1, the  G--function, the zeros of z(s) and L(s,c), the Riemann Hypothesis (a XIXth century problem for the XXIst  century mathematicians), the Prime Number Theorem, distribution of primes in arithmetic progressions.

Other prime number sums. (If time permits: The Goldbach--Vinogradov theorem, the Polya--Vinogradov inequality, the large sieve, the Bombieri--Vinogradov theorem.)