- Meissel–Mertens constant
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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).
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.
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
- The sum of the reciprocals of the primes diverges
- Prime zeta function
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
Categories:- Mathematical constants
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