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# Legendre

[luh-zhahn-der, -zhahnd; Fr. luh-zhahn-druh]
Legendre, Adrien Marie, 1752-1833, French mathematician. He is noted especially for his work on the theory of numbers, on which he wrote an essay (1798) containing the law of quadratic reciprocity as well as several supplements, all later incorporated in a definitive work, Théorie des nombres (1830). The results of his long study of elliptic integrals appeared in Traité des fonctions elliptiques (3 vol., 1825-32). He invented independently of C. F. Gauss, and was the first to state in print (1806), the method of least squares, and he collaborated in drawing up centesimal trigonometric tables. He taught at the École militaire, Paris, and at the École normale and was associated with the bureau of longitudes from 1812. His Éléments de géométrie (1794, tr. 1867) was an influential textbook.
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 $scriptstylepi\left(x\right)$. Its value is now known to be exactly 1.

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

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

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.