Editing Analytic number theory (section)

=== Modern times ===
The biggest technical change after 1950 has been the development of ''[[Sieve theory|sieve methods]]'',{{sfn|Tenenbaum|1995|p=56}} particularly in multiplicative problems. These are [[combinatorics|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]]'',{{sfn|Tenenbaum|1995|p=267}} which uses methods from probability theory to estimate the distribution of number theoretic functions, such as how many prime divisors a number has.

Specifically, the breakthroughs by [[Yitang Zhang]], [[James Maynard (mathematician)|James Maynard]], [[Terence Tao]] and [[Ben Green (mathematician)|Ben Green]] have all used the [[Daniel Goldston|Goldston]]–[[János Pintz|Pintz]]–[[Cem Yıldırım|Yıldırım]] method, which they originally used to prove that<ref>{{Cite arXiv |last=Green |first=Ben |date=2014-02-22 |title=Bounded gaps between primes |class=math.NT |eprint=1402.4849 }}</ref><ref>{{cite journal
 | last = Maynard | first = James
 | arxiv = 1604.01041
 | doi = 10.1007/s00222-019-00865-6
 | issue = 1
 | journal = Inventiones Mathematicae
 | pages = 127–218
 | title = Primes with restricted digits
 | volume = 217
 | year = 2019| bibcode = 2019InMat.217..127M
 }}</ref><ref>{{cite journal
 | last1 = Green | first1 = Ben
 | last2 = Tao | first2 = Terence
 | arxiv = math/0404188
 | doi = 10.4007/annals.2008.167.481
 | issue = 2
 | journal = Annals of Mathematics |series=2nd Series
 | pages = 481–547
 | title = The primes contain arbitrarily long arithmetic progressions
 | volume = 167
 | year = 2008}}</ref><ref>{{Cite web |title=Bounded gaps between primes - Polymath Wiki |url=https://asone.ai/polymath/index.php?title=Bounded_gaps_between_primes |access-date=2022-07-14 |website=asone.ai |archive-date=2020-12-08 |archive-url=https://web.archive.org/web/20201208045925/https://asone.ai/polymath/index.php?title=Bounded_gaps_between_primes |url-status=dead }}</ref><ref>{{Citation |title=Terence Tao - Large and Small Gaps in the Primes [2015] | date=15 December 2017 |url=https://www.youtube.com/watch?v=LikuKTZzgoU |language=en |access-date=2022-07-14}}</ref><ref name=":0" />

<math display="block">p_{n+1}-p_n \geq o(\log p_n).</math>

Developments within analytic number theory are often refinements of earlier techniques, which reduce the error terms and widen their applicability. For example, the [[Hardy–Littlewood circle method|''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 function]]s that are not [[generating function]]s—their coefficients are constructed by use of a [[pigeonhole principle]]—and involve [[several complex variables]]. The fields of Diophantine approximation and [[Transcendental element|transcendence theory]] have expanded, to the point that the techniques have been applied to the [[Mordell conjecture]].