View source for Legendre's constant

{{short description|Constant of proportionality of prime number density}}
{{log(x)}}
[[File:Legendre's constant.svg|thumb|right|250px|The first 100,000 elements of the sequence ''a<sub>n</sub>'' =&nbsp;log(''n'')&nbsp;&minus;&nbsp;''n''/{{pi}}(''n'') (red line) appear to converge to a value around 1.08366 (blue line).]]
[[File:Legendre's constant 10 000 000.svg|thumb|right|250px|Later elements up to 10,000,000 of the same sequence ''a<sub>n</sub>'' =&nbsp;log(''n'')&nbsp;&minus;&nbsp;''n''/{{pi}}(''n'') (red line) appear to be consistently less than 1.08366 (blue line).]]
'''Legendre's constant''' is a [[mathematical constant]] occurring in a formula constructed by [[Adrien-Marie Legendre]] to approximate the behavior of the [[prime-counting function]] <math>\pi(x)</math>. The value that corresponds precisely to its [[asymptotic behavior]] is now known to be&nbsp;1.

Examination of available numerical data for known values of <math>\pi(x)</math> led Legendre to an approximating formula.

Legendre proposed in 1808 the formula 
<math display="block">y=\frac{x}{\log(x) - 1.08366},</math> ({{OEIS2C|A228211}}), as giving an approximation of <math>y=\pi(x)</math> with a "very satisfying precision".<ref>{{cite book|last=Legendre|first=A.-M.|title= Essai sur la théorie des nombres|publisher=Courcier|year=1808|page=394|url=https://gallica.bnf.fr/ark:/12148/bpt6k62826k/f420.item}}</ref><ref name=P.Ribenboim>{{cite book|last=Ribenboim|first=Paulo|title=The Little Book of Bigger Primes|date=2004|publisher=Springer-Verlag|location=New York|isbn=0-387-20169-6|page=188}}</ref>

Today, one defines the real constant <math>B</math> by
<math display="block">\pi(x)\sim\frac{x}{\log(x) - B},</math>
which is solved by putting
<math display="block">B = \lim_{n \to \infty } \left( \log(n) - {n \over \pi(n)} \right),</math>
provided that this limit exists.

Not only is it now known that the limit exists, but also that its value is equal to 1, somewhat less than Legendre's {{val|1.08366|fmt=none}}. Regardless of its exact value, the existence of the limit <math>B</math> implies the [[prime number theorem]].

[[Pafnuty Chebyshev]] proved in 1849<ref>[[Edmund Landau]]. Handbuch der Lehre von der Verteilung der Primzahlen, page 17. Third (corrected) edition, two volumes in one, 1974, Chelsea 1974</ref> that if the limit ''B'' exists, it must be equal to 1. An easier proof was given by Pintz in 1980.<ref>{{Cite journal |last=Pintz |first=Janos |date=1980 |title=On Legendre's Prime Number Formula |url=https://www.jstor.org/stable/2321863 |journal=[[The American Mathematical Monthly]] |volume=87 |issue=9 |pages=733–735 |doi=10.2307/2321863 |jstor=2321863 |issn=0002-9890}}</ref>

It is an immediate consequence of  the [[prime number theorem]], under the precise form with an explicit estimate of the error term

<math display="block"> \pi(x) = \operatorname{Li} (x) + O \left(x e^{-a\sqrt{\log x}}\right) \quad\text{as } x \to \infty</math>

(for some positive constant ''a'', where ''O''(…) is the [[big O notation]]), as proved in 1899 by [[Charles Jean de la Vallée-Poussin|Charles de La Vallée Poussin]],<ref>La Vallée Poussin, C. Mém. Couronnés Acad. Roy. Belgique 59, 1–74, 1899</ref> that ''B'' indeed is equal to 1. (The prime number theorem had been proved in 1896, independently by [[Jacques Hadamard]]<ref>''Sur la distribution des zéros de la fonction <math>\zeta(s)</math> et ses conséquences arithmétiques'', Bulletin de la Société Mathématique de France, Vol. 24, 1896, pp.&nbsp;199–220 [http://www.numdam.org/numdam-bin/fitem?id=BSMF_1896__24__199_1 Online] {{webarchive|url=https://web.archive.org/web/20120717195014/http://www.numdam.org/numdam-bin/fitem?id=BSMF_1896__24__199_1 |date=2012-07-17 }}</ref> and La Vallée Poussin,<ref>« Recherches analytiques sur la théorie des nombres premiers », Annales de la société scientifique de Bruxelles, vol. 20, 1896, pp. 183–256 et 281–361</ref> but without any estimate of the involved error term).

Being evaluated to such a simple number has made the term Legendre's constant mostly only of historical value, with it often (technically incorrectly) being used to refer to Legendre's first guess 1.08366... instead.

==References==
{{reflist}}

==External links==
*{{mathworld|urlname=LegendresConstant|title=Legendre's constant}}

{{Prime number conjectures}}

[[Category:Conjectures about prime numbers]]
[[Category:Mathematical constants]]
[[Category:1 (number)]]
[[Category:Integers]]
[[Category:Analytic number theory]]