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 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. If the n axis were plotted in the linear scale instead of logarithmic, then the figure would be $5.33(3)\times 10^9$ km long —approximately the distance to Neptune.

The value of M is approximately

M ≈ 0.2614972128476427837554268386086958590516... Template: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

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