Legendre's constant

Legendre's constant is a mathematical constant occurring in a formula constructed by Adrien-Marie Legendre to approximate the behavior of the prime-counting function ${\displaystyle \pi (x)}$. The value that corresponds precisely to its asymptotic behavior is now known to be 1.

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

Later elements up to 10,000,000 of the same sequence a_n = log(n) − n/π(n) (red line) appear to be consistently less than 1.08366 (blue line).

Examination of available numerical data for known values of ${\displaystyle \pi (x)}$ led Legendre to an approximating formula.

Legendre constructed in 1808 the formula

${\displaystyle \pi (x)\approx {\frac {x}{\log(x)-B}},}$

where ${\displaystyle B=1.08366}$ (OEIS: A228211), as giving an approximation of ${\displaystyle \pi (x)}$ with a "very satisfying precision".[1][2]

Today, one defines the value of ${\displaystyle B}$ such that

${\displaystyle \pi (x)\sim {\frac {x}{\log(x)-B}},}$

which is solved by putting

${\displaystyle B=\lim _{n\to \infty }\left(\log(n)-{n \over \pi (n)}\right),}$

provided that this limit exists.

Not only is it now known that the limit exists, but also that its value is equal to ${\displaystyle 1,}$ somewhat less than Legendre's ${\displaystyle 1.08366.}$ Regardless of its exact value, the existence of the limit ${\displaystyle B}$ 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

${\displaystyle \pi (x)=\operatorname {Li} (x)+O\left(xe^{-a{\sqrt {\log x}}}\right)\quad {\text{as }}x\to \infty }$

(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.