Definition:Legendre's Constant
Definition
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:
- $\displaystyle \lim_{n \to \infty} \pi \left({n}\right) - \frac n {\log(n)} = B$
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 $1.08366$. This sequence is ?????? in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Find out whether this is in OEIS - the link's currently down so I can't check.
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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.
