Definition:Legendre's Constant
Definition
Legendre's 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:
- $\ds \lim_{n \mathop \to \infty} \map \pi n - \frac n {\map \ln n} = B$
where $B$ is what is now known as 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.
Also known as
Legendre's constant is also known as the Legendre constant.
Also see
- Results about Legendre's constant can be found here.
Source of Name
This entry was named for Adrien-Marie Legendre.
Sources
- 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $1,08366 \ldots$