In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic progressions. Another major milestone in the subject is the prime number theorem.
Analytic number theory can be split up into two major parts, divided more by the type of problems they attempt to solve than fundamental differences in technique. Multiplicative number theory deals with the distribution of the prime numbers, such as estimating the number of primes in an interval, and includes the prime number theorem and Dirichlet's theorem on primes in arithmetic progressions. Additive number theory is concerned with the additive structure of the integers, such as Goldbach's conjecture that every even number greater than 2 is the sum of two primes. One of the main results in additive number theory is the solution to Waring's problem.
Developments within analytic number theory are often refinements of earlier techniques, which reduce the error terms and widen their applicability. For example, 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 technical change after 1950 has been the development of sieve methods as a tool, particularly in multiplicative problems. These are combinatorial in nature, and quite varied. The extremal branch of combinatorial theory has in return been greatly influenced by the value placed in analytic number theory on quantitative upper and lower bounds. Another recent development is probabilistic number theory, which uses tools from probability theory to estimate the distribution of number theoretic functions, such as how many prime divisors a number has.
Other articles related to "analytic number theory, number, theory, number theory, numbers, analytic number":
... Abstract analytic number theory is a branch of mathematics which takes the ideas and techniques of classical analytic number theory and applies them to a variety of ... The classical prime number theorem serves as a prototypical example, and the emphasis is on abstract asymptotic distribution results ... The theory was invented and developed by John Knopfmacher in the early 1970s ...
... Richert made contributions to additive number theory, Dirichlet series, Riesz summability, the multiplicative analog of the Erdős–Fuchs theorem, estimates of the ...
... arithmetic implies that Euler's proof of the infinity of prime numbers makes use of the divergence of the term at the left hand side for s = 1 (the so-called harmonic series ... This was the beginning of analytic number theory ... Analytic number theorists are often interested in the error of approximations such as the prime number theorem ...
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