Editing Legendre's constant (section)

{{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|lang=fr|trans-title=Essay on number theory|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|author-link=Paulo Ribenboim |title=The Little Book of Bigger Primes|date=2004|publisher=Springer-Verlag|location=New York|isbn=0-387-20169-6|page=163}}</ref>

However, if one defines the real function <math>B(x)</math> by
<math display="block">\pi(x)=\frac{x}{\log(x) - B(x)},</math>
and if <math>B(x)</math> converges to a real constant <math>B</math> as <math>x</math> tends to infinity, then this constant satisfies
<math display="block">B = \lim_{x \to \infty } \left( \log(x) - {x \over \pi(x)} \right).</math>

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>{{cite journal
 |title=Sur la distribution des zéros de la fonction <math>\zeta(s)</math> et ses conséquences arithmétiques
 |trans-title=On the distribution of the zeros of the function <math>\zeta(s)</math> and its arithmetic consequences
 |lang=fr
 |first=Jacques |last=Hadamard |author-link=Jacques Hadamard
 |journal=[[Bulletin de la Société Mathématique de France]]
 |volume=24
 |year=1896
 |pages=199–220
 |doi=10.24033/bsmf.545 |doi-access=free
 }}</ref> and La Vallée Poussin,<ref>{{cite book
 |title=Recherches analytiques sur la théorie des nombres premiers
 |trans-title=Analytical research on prime number theory
 |lang=fr
 |first=Charles Jean |last=de la Vallée Poussin |author-link=Charles Jean de la Vallée Poussin
 |year=1897 |location=Brussels |publisher=Hayez
 |pages=183–256, 281–361
 |url=https://archive.org/details/recherchesanaly00pousgoog
}} Originally published in {{lang|fr|Annales de la société scientifique de Bruxelles}} vol. 20 (1896). [https://archive.org/details/recherchesanalyt00lava Second scanned version], from a different library.</ref>{{rp|183–256, 281–361}}{{Page needed|date=December 2024|reason=The book has 320–330 pages, depending on how you count, but is in four separately numbered parts, with page numbers ending at 74, 119, 93 and 26, respectively.  There is no page numbered 183, let alone 361.}} 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.