# Legendre's constant

The first 100,000 elements of the sequence an = ln(n) − n/π(n) (red line) appear to converge to a value around 1.08366 (blue line).

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 an approximate formula.

Legendre conjectured in 1808 that

$\pi(x) = \frac{x}{\ln(x) - B(x)}$

where $\lim_{x \to \infty} B(x) = 1.08366$....[1]

Or similarly,

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

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

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

Charles Jean de la Vallée-Poussin, who proved the prime number theorem[2] (independently from Jacques Hadamard), finally showed that B is 1. An easier proof was given by Pintz in 1980.[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.

It should be noted that Pierre Dusart proved in 2010:

$\frac {x} {\ln x - 1} < \pi(x)$ for $x \ge 5393$, and
$\pi(x) < \frac {x} {\ln x - 1.1}$ for $x \ge 60184$.[3] This is of the same form as
$\pi(x) = \frac{x}{\ln(x) - A(x)}$ with $1 \le A(x) < 1.1$.

## References

1. ^ a b Ribenboim, Paulo (2004). The Little Book of Bigger Primes. New York: Springer-Verlag. p. 188. ISBN 0-387-20169-6.
2. ^ Vallée Poussin, C. Mém. Couronnés Acad. Roy. Belgique 59, 1-74, 1899
3. ^ Dusart, Pierre. "ESTIMATES OF SOME FUNCTIONS OVER PRIMES WITHOUT R.H.". arxiv.org. Retrieved 22 April 2014.