Definition:Legendre's Constant

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Definition

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

Legendre conjectured in $1796$ that $\map \pi n$ satisfies:

$\displaystyle \lim_{n \mathop \to \infty} \map \pi n - \frac n {\map \ln n} = B$

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 $1 \cdotp 08366$. This sequence is A228211 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


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

In $1896$, Jacques Salomon Hadamard and Charles de la Vallée Poussin independently proved the Prime Number Theorem and showed that $B$ is in fact equal to $1$.

Legendre's first guess of $1 \cdotp 08366 \ldots$ 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.


Source of Name

This entry was named for Adrien-Marie Legendre.


Sources