Meissel-Mertens constant $M$
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Number
$M$
0.261497212847642783755426838608695859051566648261199206192064213924924510897368209714142631434246651051617
Definition
The Meissel-Mertens constant $M$ can be defined as the limiting difference between the harmonic series summed only over primes and the natural logarithm of the natural logarithm. $M = \lim_{n \to \infty} \left( \sum_{p \le n} \frac{1}{p} - \ln (\ln n) \right)$.
Formulas
(1)
$M = \gamma + \sum_{p} \left[ \ln \left(1 - \frac{1}{p} \right) + \frac{1}{p} \right]$, where $\gamma = \gamma_0$ is the Euler-Mascheroni constant.
(2)
$M = \lim_{n \to \infty} \left( \sum_{p \le n} \frac{1}{p} - \ln (\ln n) \right)$.