Legendre's constant
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....
Or similarly,
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.
Pafnuty Chebyshev proved in 1849 that if the limit B exists, it must be equal to 1. An easier proof was given by Pintz in 1980.
It is an immediate consequence of the prime number theorem, under the precise form with an explicit estimate of the error term
, as proved in 1899 by Charles de La Vallée Poussin, that B indeed is equal to 1..
Being evaluated to such a simple number has made the term Legendre's constant mostly only of historical value, with it often being used to refer to Legendre's first guess 1.08366... instead.
Pierre Dusart proved in 2010
