Project IV (MATH4072) 2024-25

Advanced Topics in Analytic Number Theory


P. Vishe and A. Mangerel

Description

Analytic number theory is a branch of mathematics which uses techniques from analysis to solve problems relating to integers. A central problem in number theory is to understand the distribution of prime numbers which are building blocks of all integers. One of a key tools to studying these is provided by the Sieve Theory. This is a branch of number theory and combinatorics devoted to developing general techniques for counting integers with prescribed constraints. A significant theme in the evolution of the subject is in obtaining upper and lower bounds for the number of prime numbers having special forms, e.g., the number of twin primes, or the number of primes whose decimal expansions lack the digit 3. This area is one of both classical interest as well as active research, with several spectacular applications over the last decade, including the celebrated results of Zhang and (independently) Maynard and Tao on the existence of infinite sequences of primes with bounded gaps.

The primary goal of this project is to survey some of the classical results, including:
  • the sieve of Eratosthenes-Legendre, and first estimates on the number of primes below a given bound
  • Brun's combinatorial sieve
  • Selberg's upper bound sieve method
  • the large and larger sieves, with applications to finding quadratic residues.

These should serve as an elementary jumping off point to various applications of the topic, including
  • Artin's conjecture on primitive roots, and the work of Hooley and of Heath-Brown
  • estimates for smooth/friable numbers, with relationships to factoring algorithms and complexity
  • the Bateman-Horn conjecture on prime values of polynomials
  • the Maynard-Tao sieve, towards bounded gaps between primes.
Along the way, we will cultivate some familiarity with some of the principal results and basic techniques of analytic number theory.
Prerequisites: Familiarity with Complex Analysis II, Analysis in Many Variables II and Elementary Number Theory II will be assumed. Knowledge of Analysis III (Fourier series) would be helpful. An interest in analysis and number theory is a must.

Resources:
  1. An introduction to Sieve Methods and their Applications. A. C. Cojocaru and M.R. Murty, LMS Student Texts vol. 66, 2009.
  2. Kevin Ford's UIUC Sieve Theory Notes, Spring 2023.

email: Pankaj Vishe

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