Legendre constant

The first 100,000 terms of the sequence (red) indicate a convergence to 1.08366 (blue). ${\displaystyle a_{n}=\ln(n)-{\frac {n}{\pi (n)}}}$

The Legendre's constant is a mathematical constant that in a 1798 by Adrien-Marie Legendre occurs established formula that the number of prime numbers estimates that are not greater than a given number are. Its value was later determined to be exactly 1. ${\displaystyle n}$

On the basis of his considerations on the frequency of prime numbers, Legendre assumed that the following limit exists:

${\displaystyle \lim _{n\to \infty }{\biggl (}\!\ln(n)-{\frac {n}{\pi (n)}}{\biggr )}=B,}$

where is the natural logarithm of , the number of prime numbers that are not greater than , and the Legendre constant, which Legendre estimated to be about 1.08366 with the help of calculations up to initially = 400,000, later = 1,000,000. The prime number theorem follows from the existence of the constants, regardless of their exact value . ${\displaystyle \ln(n)}$ ${\displaystyle n}$ ${\displaystyle \pi (n)}$ ${\displaystyle n}$ ${\displaystyle B}$ ${\displaystyle n}$ ${\displaystyle n}$

Pafnuti Lwowitsch Tschebyschow proved in 1849 that this limit value has the value 1, if it exists. A simple proof was published by János Pintz in 1980 . ${\displaystyle B}$

It is a direct consequence of the prime number theorem , in the following precise form by Charles de La Vallée Poussin ,

${\displaystyle \pi (x)={\rm {Li}}(x)+O\left(x\mathrm {e} ^{-a{\sqrt {\ln x}}}\right)\quad {\text{wenn }}x\to \infty }$

(for a positive constant where the Landau symbol is) that actually exists and is 1. The prime number theorem was proven in 1896 independently by Jacques Hadamard and Charles de La Vallée Poussin (without residual estimate). ${\displaystyle a}$ ${\displaystyle O(\dotso )}$ ${\displaystyle B}$

Literature and Sources

1. ^ Adrien-Marie Legendre : Essai sur la théorie des nombres , Duprat, Paris 1798, p. 19 ; 2nd edition, Courcier, Paris 1808, p. 394 ; Théorie des nombres (Volume 2), 3rd edition, Didot, Paris 1830, p. 65 (French)
2. ↑ Edmund Landau : Handbook of the theory of the distribution of prime numbers , Third (corrected) edition, two volumes in one, Chelsea 1974, p. 17.
3. ↑ János Pintz: On Legendre's prime number formula . In: American Mathematical Monthly The American Mathematical Monthly , Vol. 87 (1980), pp. 733-735.
4. ↑ La Vallée Poussin, C. Mém. Couronnés Acad. Roy. Belgique 59, 1-74, 1899
5. ↑ Sur la distribution des zéros de la fonction et ses conséquences arithmétiques ${\displaystyle \zeta (s)}$ , Bulletin de la Société Mathématique de France, Vol. 24, 1896, pp. 199–220 Online ( Memento of the original from July 17, 2012 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. 
6. ^ "Recherches analytiques sur la théorie des nombres premiers", Annales de la société scientifique de Bruxelles, vol. 20, 1896, pp. 183-256 and 281-361

Web link

• Eric W. Weisstein : Legendre's Constant . In: MathWorld (English).