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, 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:

[math]\displaystyle{ M = \lim_{n \rightarrow \infty } \left( \sum_{\scriptstyle p\text{ prime}\atop \scriptstyle p\le n} \frac{1}{p} - \ln(\ln n) \right)=\gamma + \sum_{p} \left[ \ln\! \left( 1 - \frac{1}{p} \right) + \frac{1}{p} \right]. }[/math]

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 [math]\displaystyle{ n=2^{15}, 2^{16}, \ldots, 2^{46} \approx 7.04 \times 10^{13} }[/math] 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 [math]\displaystyle{ 5.33(3) \times 10^9 }[/math] 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.

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. 
• On the remainder in a series of Mertens (postscript file)