Meissel-Mertens constant

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The Meissel-Mertens constant, also referred to as Mertens constant, Kronecker's constant, Hadamard-de la Vallée-Poussin constant or prime reciprocal constant, is a mathematical constant, used mainly in number theory, and is defined as the limiting difference between the harmonic series summed only over the primes and the natural logarithm of the natural logarithm:

$M = \lim_{n \rightarrow \infty } \left( \sum_{p \leq n} \frac{1}{p} - \ln(\ln(n)) \right)=\gamma + \sum_{p} \left[ \ln \left( 1 - \frac{1}{p} \right) + \frac{1}{p} \right]$

Its value is approximately

M ≈ 0.261497212847642783755426838608695859...

Here γ is the famous Euler-Mascheroni constant, which has a similar definition involving a sum over all integers (not just the primes).

The fact that there are two logarithms (log of a log) in the limit for the Meissel-Mertens constant may be thought of as a consequence of the combination of the prime number theorem and the limit of the Euler-Mascheroni constant.