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 %2B \sum_{p} \left[ \ln\! \left( 1 - \frac{1}{p} \right) %2B \frac{1}{p} \right].](/2012-wikipedia_en_all_nopic_01_2012/I/eebb86187e4807a42426ea2cd6d820d9.png)
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.[1]
See also
References
- ^ Reuters (July 5, 2011). "Google's strange bids for Nortel patents". FinancialPost.com. http://business.financialpost.com/2011/07/05/googles-strage-bids-for-nortel-patents/. Retrieved 2011-08-16.
External links
- Weisstein, Eric W., "Mertens Constant" from MathWorld.
- On the remainder in a series of Mertens (postscript file)s