Legendre's constant is a mathematical constant occurring in a formula conjectured by Adrien-Marie Legendre to capture the asymptotic behavior of the prime-counting function . Its value is now known to be exactly 1.
Examination of available numerical evidence for known primes led Legendre to suspect that satisfies an approximate formula.
Legendre conjectured in 1808 that
where B is Legendre's constant. He guessed B to be about 1.08366, but regardless of its exact value, the existence of B implies the prime number theorem.
It is an immediate consequence of the proof of the prime number theorem under the form
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.
Pierre Dusart proved in 2010
- for , and
- for . This is of the same form as
- with .
- Ribenboim, Paulo (2004). The Little Book of Bigger Primes. New York: Springer-Verlag. p. 188. ISBN 0-387-20169-6.
- Edmund Landau. Handbuch der Lehre von der Verteilung der Primzahlen, page 17. Third (corrected) edition, two volumes in one, 1974, Chelsea 1974
- J. Pintz. On Legendre's prime number formula. Amer. Math. Monthly 87 (1980), 733-735.
- Sur la distribution des zéros de la fonction et ses conséquences arithmétiques, Bulletin de la Société Mathématique de France, Vol. 24, 1896, pp. 199–220 Online
- La Vallée Poussin, C. Mém. Couronnés Acad. Roy. Belgique 59, 1-74, 1899
- Dusart, Pierre. "ESTIMATES OF SOME FUNCTIONS OVER PRIMES WITHOUT R.H.". arxiv.org. Retrieved 22 April 2014.