Meissel–Mertens constant

Short description: Mathematical 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 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:

$M=\underset{n\to \mathrm{\infty }}{\lim }(\sum _{\genfrac{}{}{0ex}{}{p\text{ prime}}{p\le n}}\frac{1}{p}-\ln (\ln n))=\gamma +\sum _p[\ln (1-\frac{1}{p})+\frac{1}{p}].$

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 $n=2^{15},2^{16},\dots ,2^{46}\approx 7.04\times 10^{13}$ 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 $5.33(3)\times 10^9$ km long —which is approximately the distance to Neptune.

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.

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: $1,902,160,540 (Brun's constant), $2,614,972,128 (Meissel–Mertens constant), and $3.14159 billion (π).

See also

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

External links

• Weisstein, Eric W.. "Mertens Constant". http://mathworld.wolfram.com/MertensConstant.html. 
• Lindqvist, Peter; Peetre, Jaak (2007), On the remainder in a series of Mertens, https://citeseerx.ist.psu.edu/document?repid=rep1&doi=0ff00f98eb25cbd076cf7a18f85ae51e4b95f618 
• Meissel, Ernst (1870). "Ueber die Bestimmung der Primzahlenmenge innerhalb gegebener Grenzen". Mathematische Annalen 2 (4): 636-642. doi:10.1007/BF01444045. https://eudml.org/doc/156468. 
• Mertens, Franz (1874). "Ein Beitrag zur analytischen Zahlentheorie". J. reine angew. Math. 78: 46-62. doi:10.1515/crll.1874.78.46. https://eudml.org/doc/148244.