Legendre's constant explained

. The value that corresponds precisely to its asymptotic behavior is now known to be 1.

Examination of available numerical data for known values of

led Legendre to an approximating formula.

Legendre proposed in 1808 the formula $$y=\backslash frac,$$, as giving an approximation of

with a "very satisfying precision".^{[1] [2]}

Today, one defines the real constant

by$$\backslash pi(x)\backslash sim\backslash frac,$$which is solved by putting$$B\; =\; \backslash lim\_\; \backslash left(\backslash log(n)\; -\; \backslash right),$$provided that this limit exists.

Not only is it now known that the limit exists, but also that its value is equal to 1, somewhat less than Legendre's . Regardless of its exact value, the existence of the limit

implies the prime number theorem.

Pafnuty Chebyshev proved in 1849^[3] that if the limit B exists, it must be equal to 1. An easier proof was given by Pintz in 1980.^[4]

It is an immediate consequence of the prime number theorem, under the precise form with an explicit estimate of the error term

$$\backslash pi(x)\; =\; \backslash operatorname\; (x)\; +\; O\; \backslash left(x\; e^\backslash right)\; \backslash quad\backslash text\; x\; \backslash to\; \backslash infty$$

(for some positive constant a, where O(…) is the big O notation), as proved in 1899 by Charles de La Vallée Poussin,^[5] that B indeed is equal to 1. (The prime number theorem had been proved in 1896, independently by Jacques Hadamard^[6] and La Vallée Poussin,^[7] but without any estimate of the involved error term).

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.

Notes and References

1. Book: Legendre, A.-M.. Essai sur la théorie des nombres. Courcier. 1808. 394.
2. Book: Ribenboim, Paulo. The Little Book of Bigger Primes. 2004. Springer-Verlag. New York. 0-387-20169-6. 188.
3. [Edmund Landau]
4. Pintz . Janos . 1980 . On Legendre's Prime Number Formula . . 87 . 9 . 733–735 . 10.2307/2321863 . 2321863 . 0002-9890.
5. La Vallée Poussin, C. Mém. Couronnés Acad. Roy. Belgique 59, 1–74, 1899
6. Sur la distribution des zéros de la fonction

   \zeta(s)

   et ses conséquences arithmétiques, Bulletin de la Société Mathématique de France, Vol. 24, 1896, pp. 199–220 Online
7. « Recherches analytiques sur la théorie des nombres premiers », Annales de la société scientifique de Bruxelles, vol. 20, 1896, pp. 183–256 et 281–361