Meissel–Mertens constant

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).

The Meissel–Mertens constant (named after Ernst Meissel and Franz Mertens), also referred to as Mertens constant, Kronecker's constant, Hadamardde la Vallée-Poussin constant or 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:

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

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

The plot of the prime harmonic sum up to $n=2^{15}, 2^{16}, \ldots, 2^{46}=7.04\times 10^{13}$ 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 $5.33(3)\times 10^9$ km long --- that is the size of the Solar System.

The value of M is approximately

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

Mertens' 2nd theorem says 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 number 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]