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 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 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 as a bid in the Nortel patent auction. The bid posted by Google was one of three that were based on mathematical numbers.

See also

• The sum of the reciprocals of the primes diverges
• Prime zeta function

External links

• Weisstein, Eric W., "Mertens Constant" from MathWorld.
• On the remainder in a series of Mertens (postscript file)s