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[[File:Meissel–Mertens constant definition.svg|thumb|300px|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).]]
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The '''Meissel–Mertens constant''' (named after [[Ernst Meissel]] and [[Franz Mertens]]), also referred to as '''Mertens constant''', '''[[Leopold Kronecker|Kronecker]]'s constant''', '''[[Jacques Hadamard|Hadamard]]–[[Charles Jean de la Vallée-Poussin|de la Vallée-Poussin]] constant''' or '''prime reciprocal constant''', is a [[mathematical constant]] in [[number theory]], defined as the [[limit (mathematics)|limiting]] difference between the [[harmonic series (mathematics)|harmonic series]] summed only over the [[prime number|primes]] and the [[natural logarithm]] of the natural logarithm:
 
:<math>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].</math>
 
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'' &asymp; 0.2614972128476427837554268386086958590516... {{OEIS|id=A077761}}.
 
[[Mertens' theorems|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 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 ( [[pi|π]] ).<ref name="FP_2011">{{cite news
  | author = Reuters
  | authorlink = Reuters
  | url = http://business.financialpost.com/2011/07/05/googles-strage-bids-for-nortel-patents/
  | title = Google's strange bids for Nortel patents
  | newspaper = [[Financial Post|FinancialPost.com]]
  | date = July 5, 2011
  | accessdate = 2011-08-16
}}</ref>
 
==See also==
* [[Divergence of the sum of the reciprocals of the primes]]
* [[Prime zeta function]]
 
== References ==
{{reflist}}
 
==External links==
* {{MathWorld|urlname=MertensConstant|title=Mertens Constant}}
* [http://www.maths.lth.se/matematiklu/personal/jaak/Mertens,rest.ps ''On the remainder in a series of Mertens''] (postscript file)
 
{{DEFAULTSORT:Meissel-Mertens constant}}
[[Category:Mathematical constants]]

Latest revision as of 09:07, 30 November 2014

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