Math 571 Analytic Number Theory I, Fall 2004


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Introduction

Our objective, starting only from the most elementary considerations, is to study a range of important number theoretic questions by the use of analytic techniques.  For example, Hermann Weyl's seminal paper of 1917 on uniform distribution is

Hermann Klaus Hugo Weyl

Hermann Klaus Hugo Weyl, 1885-1955

dependent on very simple ideas from harmonic analysis, which we will develop from scratch in the course.  This paper is one of the highlights of 20th century mathematics and has been very influential.
 
Following the famous work of Hardy and Ramanujan on the partition function, and using ideas from Weyl's paper, in the 1920s Hardy and Littewood introduced a

G H Hardy                         J E Littlewood                         S Ramanujan

G. H. Hardy, 1877-1947, J. E. Littlewood, 1885-1977, S. Ramanujan, 1887-1920  

general method for dealing with additive problems.  An example is Waring's problem

Edward Waring

Edward Waring, 1736-1798

which is concerned with the value of the smallest s such that for every n the equation

x_1^k + x_2^k + ...+ x_s^k = n

has a solution in non-negative integers x_i.

Another is to show that every large odd number is the sum of three prime numbers (the ternary Goldbach (1690-1764) problem).  This was established by Vinogradov

Vinogradov

Ivan Matveevich Vinogradov, 1891-1983

in 1937 by an adaptation of the Hardy-Littlewood method.

A related open question is whether every even number is prime or the sum of two primes (the binary Goldbach problem), and an associated question is whether there are infinitely many primes p such that p + 2 is also prime (the twin prime problem).  Some of the most interesting attacks on these problems have been via sieve theory.  The original sieve is that of Erathosthenes   

Erathosthenes

Eratosthenes of Cyrene, 276BC-194BC

which was designed to create a list of primes, but which can be adapted to study other questions involving prime numbers.  Modern forms of the sieve are due to Brun (1885-1978) and Selberg.

Atle Selberg

Atle Selberg, 1917-


Fundamental to many of the analytic methods in number theory are questions as to how closely a given real number can be approximated by a rational number with denominator not exceeding a given quantity, and generalisations of this are related to Minkowski's theorem in the geometry of numbers.

Minkowski

Hermann Minkowski, 1864-1909

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