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 $scriptstylepi(x)$. Its value is now known to be exactly 1.

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

:$lim\_\{n\; ightarrow\; infty\; \}\; ln(n)\; -\; \{n\; over\; pi(n)\}\; =\; B$

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.

External links

*mathworld|urlname=LegendresConstant|title=Legendre's constant