Definition:Legendre's Constant

Legendre's constant (or the Legendre constant) is a mathematical constant conjectured by Adrien-Marie Legendre to specify the prime-counting function $$\pi \left({n}\right)$$.

Legendre conjectured in 1796 that $$\pi \left({n}\right)$$ satisfies:
 * $$\lim_{n \to \infty} \pi \left({n}\right) - \frac{n}{\log(n)} = B \ $$

where $$B$$ is Legendre's constant.

If such a number $$B$$ exists, then this implies the Prime Number Theorem.

Legendre's guess for $$B$$ was about $$1.08366$$.

Later, Gauss looked at this problem and thought that $$B$$ might actually be lower.

In 1896, Hadamard and Poussin independently who proved the Prime Number Theorem and showed that $$B$$ is in fact equal to $$1$$.

Legendre's first guess of $$1.08366 \ldots$$ is still (incorrectly) referred to as Legendre's constant, even though its "correct" value is in fact exactly $$1$$.

Hence it is only now of historical importance.