The MeisselâÂÂMertens constant (named after Ernst Meissel and Franz Mertens), also referred to as the Mertens constant, Kronecker's constant (after Leopold Kronecker), HadamardâÂÂde la Vallée-Poussin constant (after Jacques Hadamard and Charles Jean de la Vallée-Poussin), 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 ó is the EulerâÂÂMascheroni constant, which has an analogous definition involving a sum over all integers (not just the primes).
The value of M is approximately
Mertens' second theorem establishes that the limit exists.
The fact that there are two logarithms (log of a log) in the limit for the MeisselâÂÂMertens constant may be thought of as a consequence of the combination of the prime number theorem and the limit of the EulerâÂÂMascheroni constant.
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 (Brun's constant), $2,614,972,128 (MeisselâÂÂMertens constant), and $3.14159 billion (ÃÂ).