Meissel–Mertens constant

The Meissel–Mertens constant (named after Ernst Meissel and Franz Mertens), also referred to as the Mertens constant, Kronecker's constant (after Leopold Kronecker), 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 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:

${\displaystyle M=\lim _{n\rightarrow \infty }\left(\sum _{\scriptstyle p{\text{ prime}} \atop \scriptstyle p\leq n}{\frac {1}{p}}-\ln(\ln n)\right)=\gamma +\sum _{p}\left[\ln \!\left(1-{\frac {1}{p}}\right)+{\frac {1}{p}}\right].}$

Here γ is the Euler–Mascheroni constant, which has an analogous definition involving a sum over all integers (not just the primes).

The value of M is approximately

M ≈ 0.2614972128476427837554268386086958590516... (sequence A077761 in the OEIS).

Mertens' second theorem establishes that the limit exists.

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.

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 (Meissel–Mertens constant), and $3.14159 billion (π).^[1]

• Divergence of the sum of the reciprocals of the primes
• Prime zeta function

• Weisstein, Eric W. "Mertens Constant". MathWorld.
• Lindqvist, Peter; Peetre, Jaak (2007), On the remainder in a series of Mertens, S2CID 18358425
• Meissel, Ernst (1870). "Ueber die Bestimmung der Primzahlenmenge innerhalb gegebener Grenzen". Mathematische Annalen. 2 (4): 636–642. doi:10.1007/BF01444045.
• Mertens, Franz (1874). "Ein Beitrag zur analytischen Zahlentheorie". J. reine angew. Math. 78: 46–62. doi:10.1515/crll.1874.78.46.