Editing Analytic number theory (section)

==History==

===Precursors===
Much of analytic number theory was inspired by the [[prime number theorem]]. Let π(''x'') be the [[prime-counting function]] that gives the number of primes less than or equal to ''x'', for any real number&nbsp;''x''. For example, π(10)&nbsp;=&nbsp;4 because there are four prime numbers (2, 3, 5 and 7) less than or equal to 10. The prime number theorem then states that ''x'' / ln(''x'') is a good approximation to π(''x''), in the sense that the [[limit of a function|limit]] of the ''quotient'' of the two functions π(''x'') and ''x'' / ln(''x'') as ''x'' approaches infinity is 1:

: <math>\lim_{x\to\infty}\frac{\pi(x)}{x/\ln(x)}=1,</math>

known as the asymptotic law of distribution of prime numbers.

[[Adrien-Marie Legendre]] conjectured in 1797 or 1798 that π(''a'') is approximated by the function ''a''/(''A'' ln(''a'')&nbsp;+&nbsp;''B''), where ''A'' and ''B'' are unspecified constants. In the second edition of his book on number theory (1808) he then made a more precise conjecture, with ''A''&nbsp;=&nbsp;1 and ''B''&nbsp;≈&nbsp;&minus;1.08366. [[Carl Friedrich Gauss]] considered the same question: "Im Jahr 1792 oder 1793" ('in the year 1792 or 1793'), according to his own recollection nearly sixty years later in a letter to Encke (1849), he  wrote in his logarithm table (he was then 15 or 16) the short note "Primzahlen unter <math>a(=\infty) \frac a{\ln a}</math>" ('prime numbers under <math>a(=\infty) \frac a{\ln a}</math>'). But Gauss never published this conjecture. In 1838 [[Peter Gustav Lejeune Dirichlet]] came up with his own approximating function,  the [[logarithmic integral]] li(''x'') (under the slightly different form of a series, which he communicated to Gauss). Both Legendre's and Dirichlet's formulas imply the same conjectured asymptotic equivalence of π(''x'') and ''x''&nbsp;/&nbsp;ln(''x'') stated above, although it turned out that Dirichlet's approximation is considerably better if one considers the differences instead of quotients.

===Dirichlet===
{{main|Johann Peter Gustav Lejeune Dirichlet}}
[[Johann Peter Gustav Lejeune Dirichlet]] is credited with the creation of analytic number theory,<ref name=Princeton>{{cite book| last = Gowers| first = Timothy |author1-link=Timothy Gowers|author2-link = June Barrow-Green| author3-link = Imre Leader|author2=June Barrow-Green |author3=Imre Leader | title=The Princeton companion to mathematics| url = https://archive.org/details/princetoncompanio00gowe| year=2008| publisher=Princeton University Press| isbn= 978-0-691-11880-2| pages= 764–765}}</ref> a field in which he found several deep results and in proving them introduced some fundamental tools, many of which were later named after him. In 1837 he published [[Dirichlet's theorem on arithmetic progressions]], using [[mathematical analysis]] concepts to tackle an algebraic problem and thus creating the branch of analytic number theory. In proving the theorem, he introduced the [[Dirichlet character]]s and [[Dirichlet L-function|L-functions]].<ref name=Princeton/><ref name=Kanemitsu>{{cite book| last = Kanemitsu| first = Shigeru|author2=Chaohua Jia| title=Number theoretic methods: future trends | year=2002| publisher=Springer| isbn= 978-1-4020-1080-4| pages= 271–274}}</ref> In 1841 he generalized his arithmetic progressions theorem from integers to the [[Ring (mathematics)|ring]] of [[Gaussian integer]]s <math>\mathbb{Z}[i]</math>.<ref name=Elstrodt>{{cite journal | last = Elstrodt | first = Jürgen | journal = Clay Mathematics Proceedings | title = The Life and Work of Gustav Lejeune Dirichlet (1805–1859) | year = 2007 | url = http://www.uni-math.gwdg.de/tschinkel/gauss-dirichlet/elstrodt-new.pdf | access-date = 2007-12-25 | archive-date = 2008-03-07 | archive-url = https://web.archive.org/web/20080307174514/http://www.uni-math.gwdg.de/tschinkel/gauss-dirichlet/elstrodt-new.pdf | url-status = dead }}</ref>

===Chebyshev===
{{main|Pafnuty Chebyshev}}
In two papers from 1848 and 1850, the Russian mathematician [[Pafnuty Chebyshev|Pafnuty L'vovich Chebyshev]] attempted to prove the asymptotic law of distribution of prime numbers. His work is notable for the use of the zeta function ζ(''s'') (for real values of the argument "s", as are works of [[Leonhard Euler]], as early as 1737) predating Riemann's celebrated memoir of 1859, and he succeeded in proving a slightly weaker form of the asymptotic law, namely, that if the limit of π(''x'')/(''x''/ln(''x'')) as ''x'' goes to infinity exists at all, then it is necessarily equal to one.<ref>{{cite journal |author=N. Costa Pereira |jstor=2322510 |title=A Short Proof of Chebyshev's Theorem |journal=American Mathematical Monthly|date=August–September 1985|pages=494–495|volume=92|doi=10.2307/2322510|issue=7}}</ref> He was able to prove unconditionally that this ratio is bounded above and below by two explicitly given constants near to 1 for all ''x''.<ref>{{cite journal |author=M. Nair |jstor=2320934 |title=On Chebyshev-Type Inequalities for Primes |journal=American Mathematical Monthly |date=February 1982 |pages=126–129 |volume=89  |doi=10.2307/2320934 |issue=2}}</ref> Although Chebyshev's paper did not prove the Prime Number Theorem, his estimates for π(''x'') were strong enough for him to prove [[Bertrand's postulate]] that there exists a prime number between ''n'' and 2''n'' for any integer ''n''&nbsp;≥&nbsp;2.

===Riemann===
{{main|Bernhard Riemann}}
{{quote box
| align = right
| width = 30%
| quote = "{{lang|de|…es ist sehr wahrscheinlich, dass alle Wurzeln reell sind. Hiervon wäre allerdings ein strenger Beweis zu wünschen; ich habe indess die Aufsuchung desselben nach einigen flüchtigen vergeblichen Versuchen vorläufig bei Seite gelassen, da er für den nächsten Zweck meiner Untersuchung entbehrlich schien.}}"<br /><br />"…it is very probable that all roots are real. Of course one would wish for a rigorous proof here; I have for the time being, after some fleeting vain attempts, provisionally put aside the search for this, as it appears dispensable for the next objective of my investigation."
| source = Riemann's statement of the Riemann hypothesis, from his 1859 paper.<ref name="Riemann1859">{{citation|first=Bernhard |last=Riemann |author-link=Bernhard Riemann |url=http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/ |title={{sic|hide=y|Ueber}} die Anzahl der Primzahlen unter einer gegebenen {{sic|hide=y|Grösse}} |year=1859 |journal=Monatsberichte der Berliner Akademie }}. In ''Gesammelte Werke'', Teubner, Leipzig (1892), Reprinted by Dover, New York (1953). [http://www.claymath.org/millennium/Riemann_Hypothesis/1859_manuscript/ Original manuscript] {{webarchive |url=https://web.archive.org/web/20130523061451/http://www.claymath.org/millennium/Riemann_Hypothesis/1859_manuscript/ |date=May 23, 2013 }} (with English translation). Reprinted in {{harv|Borwein|Choi|Rooney|Weirathmueller|2008}} and {{harv|Edwards|1974}}</ref> (He was discussing a version of the zeta function, modified so that its roots are real rather than on the critical line. See, Riemann Xi Function.)
}}

[[Bernhard Riemann]] made some famous contributions to modern analytic number theory. In [[On the Number of Primes Less Than a Given Magnitude|a single short paper]] (the only one he published on the subject of number theory), he investigated the [[Riemann zeta function]] and established its importance for understanding the distribution of [[prime numbers]]. He made a series of conjectures about properties of the [[Riemann zeta function|zeta function]], one of which is the well-known [[Riemann hypothesis]].

===Hadamard and de la Vallée-Poussin===
{{main|Jacques Hadamard|Charles Jean de la Vallée-Poussin}}
Extending the ideas of Riemann, two proofs of the [[prime number theorem]] were obtained independently by [[Jacques Hadamard]] and [[Charles Jean de la Vallée-Poussin]] and appeared in the same year (1896). Both proofs used methods from complex analysis, establishing as a main step of the proof that the Riemann zeta function ζ(''s'') is non-zero for all complex values of the variable ''s'' that have the form ''s''&nbsp;=&nbsp;1&nbsp;+&nbsp;''it'' with ''t''&nbsp;>&nbsp;0.<ref>{{cite book |last = Ingham |first = A.E. |title = The Distribution of Prime Numbers |publisher = Cambridge University Press| year = 1990 |pages = 2–5 |isbn = 0-521-39789-8}}</ref>

=== 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]].