Revision as of 20:19, 7 November 2015

Legendre's constant is denoted as $B$, where $$B= \displaystyle\lim_{n \rightarrow \infty} \left( \log(n) - \dfrac{n}{\pi(n)} \right)=1,$$ where $\log$ denotes the logarithm and $\pi$ denotes the prime counting function.

It was shown that if $B$ exists, then the prime number theorem follows from it. Legendre himself guessed $B$ to be around $1.08366$, but Chebyshev proved in $1849$ that $B=1$.

References

Legendre's constant on Wikipedia
[1]