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Analytic number theory

In number theory, analytic number theory is the branch using methods from mathematical analysis. Initially the paradigm was to apply complex analysis, as was done in the proofs of the Prime number theorem based on the Riemann zeta function.

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 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 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. 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 quantitative upper and lower bounds.

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