In mathematics, the Meissel–Mertens constant, named after Ernst Meissel and Franz Mertens, also referred to as the Mertens constant, Kronecker's constant (after Leopold Kronecker), the Hadamard–de la Vallée-Poussin constant (after Jacques Hadamard and Charles Jean de la Vallée-Poussin), or the prime reciprocal constant, is a mathematical constant that arises in number theory. It is defined as the limiting difference between the harmonic series summed over the primes and the natural logarithm of the natural logarithm:
where is the Euler–Mascheroni constant. Mertens' second theorem establishes that the limit exists. The value of is approximately .
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.
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: first $1,902,160,540 (Brun's constant), then $2,614,972,128 (the Meissel–Mertens constant), and lastly $3.14159 billion ().
See also Divergence of the sum of the reciprocals of the primes Prime zeta function
References
External links