Definition:Meissel-Mertens Constant

Definition
Consider the expression:
 * $\ds M = \map {\lim_{n \mathop \to \infty} } {\sum_{\substack {p \mathop \le n \\ \text {$p$ prime} } } \dfrac 1 p - \ln \ln n}$

Then:
 * $M \approx 0 \cdotp 26149 \, 72128 \, 47642 \, 78375 \, 54268 \, 38608 \, 69585 \, 90516 \ldots$

The constant $M$ is known as the Meissel-Mertens Constant.

Also known as
Also known as:
 * the Mertens constant
 * Kronecker's Constant, for
 * the Hadamard-de la Vallée-Poussin Constant, for and
 * the Prime Reciprocal Constant

Also see

 * Mertens' Second Theorem, where $M$ is proved to exist.