http://www.exampleproblems.com/wiki/index.php?title=Analytic_number_theory&feed=atom&action=historyAnalytic number theory - Revision history2021-04-21T11:55:29ZRevision history for this page on the wikiMediaWiki 1.33.0http://www.exampleproblems.com/wiki/index.php?title=Analytic_number_theory&diff=20448&oldid=prevTodd at 11:55, 21 April 20212021-04-21T11:55:29Z<p></p>
<p><b>New page</b></p><div>'''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|the existence of infinitely many primes in any arithmetic progression]]. The proofs of the [[prime number theorem]] based on the [[Riemann zeta function]] is another milestone.<br />
<br />
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 number]]s, 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]].<br />
<br />
Methods have changed somewhat. The ''circle method'' of [[G. H. Hardy|Hardy]] and [[John Edensor Littlewood|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 function]]s - their coefficients are constructed by use of a [[pigeonhole principle]] - and involve [[several complex variables]].<br />
The fields of diophantine approximation and [[transcendence (mathematics)|transcendence theory]] have expanded, to the point that the techniques have been applied to the [[Mordell conjecture]].<br />
<br />
The biggest single technical change after 1950 has been the development of ''[[sieve method]]s'' as an auxiliary tool, particularly in multiplicative problems. These are [[combinatorics|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.<br />
<br />
[[Category:Analytic number theory]]<br />
[[de:Analytische Zahlentheorie]]<br />
[[fr:Théorie analytique des nombres]]<br />
[[sv:Analytisk talteori]]<br />
[[zh:解析数论]]</div>Todd