Editing Meissel–Mertens constant

[[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).]]
The '''Meissel–Mertens constant''' (named after [[Ernst Meissel]] and [[Franz Mertens]]), also referred to as the '''Mertens constant''', '''Kronecker's constant''' (after [[Leopold Kronecker]]), '''Hadamard–de la Vallée-Poussin constant''' (after [[Jacques Hadamard]] and [[Charles Jean de la Vallée-Poussin]]), or the '''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_{\scriptstyle p\text{ prime}\atop \scriptstyle p\le 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 [[Euler–Mascheroni constant]], which has an analogous definition involving a sum over all integers (not just the primes).
[[File:Primes harmonic.png|thumb|300px| The plot of the prime harmonic sum up to <math>n=2^{15}, 2^{16}, \ldots, 2^{46} \approx 7.04 \times 10^{13}</math> and the Merten's approximation to it. The original of this figure has y axis of the length 8 cm and spans the interval (2.5, 3.8), so if the n axis would be plotted in the linear scale instead of logarithmic, then it should be <math>5.33(3) \times 10^9</math> km long — that is the size of the Solar System.]]

The value of ''M'' is approximately
:''M'' ≈ 0.2614972128476427837554268386086958590516... {{OEIS|id=A077761}}.

[[Mertens' theorems|Mertens' second theorem]] establishes 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 Meissel-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 ([[pi|π]]).<ref name="FP_2011">{{cite news
  | agency = 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}}
* {{Citation|first1=Peter|last1=Lindqvist|first2=Jaak|last2=Peetre|url=https://citeseerx.ist.psu.edu/document?repid=rep1&doi=0ff00f98eb25cbd076cf7a18f85ae51e4b95f618|title=On the remainder in a series of Mertens|year=2007|s2cid=18358425}}
* {{cite journal|first1=Ernst|last1=Meissel|title=Ueber die Bestimmung der Primzahlenmenge innerhalb gegebener Grenzen|year=1870|journal=Mathematische Annalen|volume=2|number=4|pages=636-642|url=https://eudml.org/doc/156468|doi=10.1007/BF01444045}}
* {{cite journal|first1=Franz|last1=Mertens|title=Ein Beitrag zur analytischen Zahlentheorie|journal=J. reine angew. Mathem. |year=1874|volume=78|pages=46-62|doi=10.1515/crll.1874.78.46|url=https://eudml.org/doc/148244|url-access=subscription}}

{{DEFAULTSORT:Meissel-Mertens constant}}
[[Category:Mathematical constants]]