Legendre's constant

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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 $\scriptstyle\pi(x)$ . Its value is now known to be exactly 1.

Examination of available numerical evidence for known primes led Legendre to suspect that $\scriptstyle\pi(x)$ satisfies:

$\lim_{n \rightarrow \infty } \ln(n) - {n \over \pi(n)} = B$

where B is Legendre's constant. He guessed B to be about 1.08366, but regardless of its exact value, B existing implies the prime number theorem.

Later Carl Friedrich Gauss also examined the numerical evidence and concluded that the limit might be lower.

Charles Jean de la Vallée-Poussin, who proved the prime number theorem (independently from Jacques Hadamard), finally showed that B is 1.

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.