The MeisselâMertens constant (named after Ernst Meissel and Franz Mertens), also referred to as Mertens constant, Kronecker's constant, Hadamardâ de la VallĂ©e-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 Îł is the EulerâMascheroni 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 MeisselâMertens constant may be thought of as a consequence of the combination of the prime number theorem and the limit of the EulerâMascheroni 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 (MeisselâMertens constant), and $3.14159 billion ( Ï). [1]
The MeisselâMertens constant (named after Ernst Meissel and Franz Mertens), also referred to as Mertens constant, Kronecker's constant, Hadamardâ de la VallĂ©e-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 Îł is the EulerâMascheroni 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 MeisselâMertens constant may be thought of as a consequence of the combination of the prime number theorem and the limit of the EulerâMascheroni 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 (MeisselâMertens constant), and $3.14159 billion ( Ï). [1]