Legendre's constant
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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 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> (Template:OEIS2C), as giving an approximation of <math>y=\pi(x)</math> with a "very satisfying precision".<ref>Template:Cite book</ref><ref name=P.Ribenboim>Template:Cite book</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 Template:Val. 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>Template:Cite journal</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 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>Template:Cite journal</ref> and La Vallée Poussin,<ref>Template:Cite book Originally published in {{#invoke:Lang|lang}} vol. 20 (1896). Second scanned version, from a different library.</ref>Template:RpTemplate:Page needed 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.
Numerical valuesEdit
Using known values for <math>\pi(x)</math>, we can compute <math>B(x) = \log x - \frac{x}{\pi(x)}</math> for values of <math>x</math> far beyond what was available to Legendre:
Values up to <math>\pi(10^{29})</math> (the first two columns) are known exactly; the values in the third and fourth columns are estimated using the Riemann R function.