Analytic number theory

Analytic number theory is the branch of number theory that uses methods from mathematical analysis. Its first major success was Dirichlet's application of analysis to prove Dirichlet's theorem on arithmetic progressions, stating the existence of infinitely many primes in arithmetic progressions of the form a + nb, where a and b are relatively prime. The proofs of the prime number theorem based on the Riemann zeta function are another milestone.

The outline of the subject remains similar to the heyday of the subject in the 1930s. Multiplicative number theory deals with the distribution of the prime numbers, applying Dirichlet series as generating functions. It is assumed that the methods will eventually apply to the general L-function, though that theory is still largely conjectural. Additive number theory has as typical problems Goldbach's conjecture and Waring's problem.

Methods have changed somewhat. The circle method of Hardy and Littlewood was conceived as applying to power series near the unit circle in the complex plane; it is now thought of in terms of finite exponential sums (that is, on the unit circle, but with the power series truncated). The needs of diophantine approximation are for auxiliary functions that aren't generating functions - their coefficients are constructed by use of a pigeonhole principle - and involve several complex variables. The fields of diophantine approximation and transcendence theory have expanded, to the point that the techniques have been applied to the Mordell conjecture.

The biggest single technical change after 1950 has been the development of sieve methods as an auxiliary tool, particularly in multiplicative problems. These are combinatorial in nature, and quite varied. Also much cited are uses of probabilistic number theory — forms of random distribution assertions on the primes, for example: these have not received any definitive shape. The extremal branch of combinatorial theory has in return been much influenced by the value placed in analytic number theory on (often separate) quantitative upper and lower bounds.

The simplest introductory book on the subject is T. M. Apostol, Introduction to analytic number theory. At the next level, the works of G. Tenenbaum, Introduction to Analytic Number Theory , and H.Davenport, Multiplicative Number Theory, 3rd. edn.,and H.L. Montgomery and R.C.Vaughan, Multiplicative Number Theory I : Classical Theory, are helpful. The most ambitious single-volume treatment is H. Iwaniec and E. Kowalski, Analytic Number Theory. On specialized aspects the following books have become especially well-known: E.C. Titchmarsh, The theory of the Riemann zeta-function, 2nd.edn. ; H.Halberstam and H.-E. Richert, Sieve Methods; and R.C. Vaughan, The Hardy-Littlewood method, 2nd. edn. Certain topics have not yet reached book form in any depth; examples are the pair-correlation conjecture (H.L. Montgomery) and the work that initiated from it; the new results of Goldston, Pintz and Yilidrim on small gaps between primes; and the recent work of B. Green and T. Tao showing that arbitrarily long arithmetic progressions of primes exist. New work is reviewed on the American Mathematical Society's MathSciNet under sections 11L,11M, 11N and 11P in particular. The site is accessible without subscription at computers located in university mathematics departments.

Number Theory