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.
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.
It should be noted that 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.
- 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.