Meissel–Mertens constant
![](../I/m/Meissel%E2%80%93Mertens_constant_definition.svg.png)
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:
Here γ is the famous Euler–Mascheroni constant, which has a similar definition involving a sum over all integers (not just the primes).
![](../I/m/Primes_harmonic.png)
![n=2^{15}, 2^{16}, \ldots, 2^{46}=7.04 \times 10^{13}](../I/m/38c5ba94e2b9d36f15ca4e28f5e7ab51.png)
![5.33(3) \times 10^9](../I/m/68b709a7c4089f65e712d3cd30000a65.png)
The value of M is approximately
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.
In popular culture
The Meisel-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).[1]
See also
References
- ↑ Reuters (July 5, 2011). "Google's strange bids for Nortel patents". FinancialPost.com. Retrieved 2011-08-16.
External links
- Weisstein, Eric W., "Mertens Constant", MathWorld.
- On the remainder in a series of Mertens (postscript file)