Legendre's constant

Legendre's constant is a "phantom" that doesn't really exist.

Before the discovery of the prime number theorem, examination of available numerical evidence for known primes had led Legendre to conjecture that perhaps

<math>\pi(n) = {n \over \ln(n) - A(n)}<math>

where

<math>\lim_{n \rightarrow \infty } A(n) \approx 1.08366...<math>.

The quantity 1.08366... was called Legendre's constant. Later Gauss also examined the numerical evidence and concluded that the limit might be lower. In fact, the best limit value of A(n) turns out to be 1. Thus, there is no "Legendre's constant".