# 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 $\map \pi n$.

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

- 1983: François Le Lionnais and Jean Brette:
*Les Nombres Remarquables*... (previous) ... (next): $1,08366 \ldots$