# Legendre's constant

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 ${\displaystyle \pi (x)}$. The value that corresponds precisely to its asymptotic behavior is now known to be 1.

Examination of available numerical data for known values of ${\displaystyle \pi (x)}$ led Legendre to an approximating formula.

Legendre constructed in 1808 the formula

${\displaystyle \pi (x)\approx {\frac {x}{\log(x)-B}},}$

where ${\displaystyle B=1.08366}$ (), as giving an approximation of ${\displaystyle \pi (x)}$ with a "very satisfying precision".[1][2]

Today, one defines the value of ${\displaystyle B}$ such that

${\displaystyle \pi (x)\sim {\frac {x}{\log(x)-B}},}$

which is solved by putting

${\displaystyle B=\lim _{n\to \infty }\left(\log(n)-{n \over \pi (n)}\right),}$

provided that this limit exists.

Not only is it now known that the limit exists, but also that its value is equal to ${\displaystyle 1,}$ somewhat less than Legendre's ${\displaystyle 1.08366.}$ Regardless of its exact value, the existence of the limit ${\displaystyle B}$ implies the prime number theorem.

Pafnuty Chebyshev proved in 1849[3] that if the limit B exists, it must be equal to 1. An easier proof was given by Pintz in 1980.[4]

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

${\displaystyle \pi (x)=\operatorname {Li} (x)+O\left(xe^{-a{\sqrt {\log x}}}\right)\quad {\text{as }}x\to \infty }$

(for some positive constant a, where O(…) is the big O notation), as proved in 1899 by Charles de La Vallée Poussin,[5] that B indeed is equal to 1. (The prime number theorem had been proved in 1896, independently by Jacques Hadamard[6] and La Vallée Poussin,[7] 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

1. ^ Legendre, A.-M. (1808). Essai sur la théorie des nombres. Courcier. p. 394.
2. ^ Ribenboim, Paulo (2004). The Little Book of Bigger Primes. New York: Springer-Verlag. p. 188. ISBN 0-387-20169-6.
3. ^ Edmund Landau. Handbuch der Lehre von der Verteilung der Primzahlen, page 17. Third (corrected) edition, two volumes in one, 1974, Chelsea 1974
4. ^ Pintz, Janos (1980). "On Legendre's Prime Number Formula". The American Mathematical Monthly. 87 (9): 733–735. doi:10.2307/2321863. ISSN 0002-9890. JSTOR 2321863.
5. ^ La Vallée Poussin, C. Mém. Couronnés Acad. Roy. Belgique 59, 1–74, 1899
6. ^ Sur la distribution des zéros de la fonction ${\displaystyle \zeta (s)}$ et ses conséquences arithmétiques, Bulletin de la Société Mathématique de France, Vol. 24, 1896, pp. 199–220 Online Archived 2012-07-17 at the Wayback Machine
7. ^ « Recherches analytiques sur la théorie des nombres premiers », Annales de la société scientifique de Bruxelles, vol. 20, 1896, pp. 183–256 et 281–361