{{Short description|Mathematical constant}} 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 limiting difference between the harmonic series summed only over the 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). 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. If the n axis were plotted in the linear scale instead of logarithmic, then the figure would be <math>5.33(3) \times 10^9</math> km long —approximately the distance to Neptune.
The value of ''M'' is approximately :''M'' ≈ 0.2614972128476427837554268386086958590516... {{OEIS|id=A077761}}.
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 (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 = 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. Math. |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