Meissel–Mertens constant


The Meissel–Mertens constant, also referred to as 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.

In popular culture

The Meissel-Mertens constant was used by Google when bidding in the Nortel patent auction. Google posted three bids based on mathematical numbers: $1,902,160,540, $2,614,972,128, and $3.14159 billion.