Math 571 Analytic Number Theory I

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

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

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

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

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

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, 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.

Hermann Minkowski, 1864-1909

TOPICS

The approximation of real numbers by rationals, diophantine approximation, criteria for irrationality, examples of transcendental numbers.
Basic harmonic analysis, with particular reference to questions in number theory. The Poisson summation formula.
The uniform distribution of sequences. The Weyl criterion. Examples, such as the sequence xn^2 when x is irrational.
Waring's problem and the simplest form of the Hardy-Littlewood method.
The Goldbach ternary problem.
Elementary sieve theory.
The large sieve, Selberg's lambda-squared sieve and Gallagher's sieve.

The Hardy-Littlewood Method (Cambridge Tracts in Mathematics, No 125) by R. C. Vaughan, Cambridge University Press, ISBN 0521573475.

Uniform distribution of sequences by L. Kuipers and H. Niederreiter, Wiley-Interscience, ISBN 0471510459.
Applications of Sieve Methods to the Theory of Numbers (Cambridge Tracts in Mathematics, No 70) by C. Hooley, Cambridge University Press, ISBN 0521209153. See also Sieves in Number Theory (Ergebnisse Der Mathematik Und Ihrer Grenzgebiete, 3. Folge, Bd. 43.) by G. R. H. Greaves, ISBN 3540416471.
An Introduction to the Geometry of Numbers (Classics in Mathematics) by J. W. S. Cassels, ISBN 3540617884.