Meissel–Mertens constant
The Meissel–Mertens constant, also referred to as the Mertens constant, Kronecker's constant, Hadamard–de la Vallée-Poussin constant, or the prime reciprocal constant, is a mathematical constant in number theory, defined as the limiting difference between the harmonic series summed only over the primes and the natural logarithm of the natural logarithm:
Here γ is the Euler–Mascheroni constant, which has an analogous definition involving a sum over all integers.
The value of M is approximately
Mertens' second theorem establishes that the limit exists.
The fact that there are two logarithms 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.