Meissel–Mertens constant

The Meissel–Mertens constant (named after Ernst Meissel and Franz Mertens), 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:

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

In the limit, the sum of the reciprocals of the primes < n and the function ln(ln n) are separated by a constant, the Meissel–Mertens constant (labelled M above).

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

The plot of the prime harmonic sum up to ${\displaystyle n=2^{15},2^{16},\ldots ,2^{46}\approx 7.04\times 10^{13}}$ and the Merten's approximation to it. The original of this figure has y axis of the length 8 cm and spans the interval (2.5, 3.8), so if the n axis would be plotted in the linear scale instead of logarithmic, then it should be ${\displaystyle 5.33(3)\times 10^{9}}$ km long — that is the size of the Solar System.

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.