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%I #96 Jul 25 2023 02:49:22
%S 2,6,1,4,9,7,2,1,2,8,4,7,6,4,2,7,8,3,7,5,5,4,2,6,8,3,8,6,0,8,6,9,5,8,
%T 5,9,0,5,1,5,6,6,6,4,8,2,6,1,1,9,9,2,0,6,1,9,2,0,6,4,2,1,3,9,2,4,9,2,
%U 4,5,1,0,8,9,7,3,6,8,2,0,9,7,1,4,1,4,2,6,3,1,4,3,4,2,4,6,6,5,1,0,5,1,6,1,7
%N Decimal expansion of Mertens's constant, which is the limit of (Sum_{i=1..k} 1/prime(i)) - log(log(prime(k))) as k goes to infinity, where prime(i) is the i-th prime number.
%C Graham, Knuth & Patashnik incorrectly give this constant as 0.261972128. - _Robert G. Wilson v_, Dec 02 2005 [This was corrected in the second edition (1994). - _T. D. Noe_, Mar 11 2017]
%C Also the average deviation of the number of distinct prime factors: sum_{n < x} omega(n) = x log log x + B_1 x + O(x) where B_1 is this constant, see (e.g.) Hardy & Wright. - _Charles R Greathouse IV_, Mar 05 2021
%C Named after the Polish mathematician Franz Mertens (1840-1927). Sometimes called Meissel-Mertens constant, after Mertens and the German astronomer Ernst Meissel (1826-1895). - _Amiram Eldar_, Jun 16 2021
%D Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2004, pp. 94-98
%D Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, A Foundation For Computer Science, Addison-Wesley, Reading, MA, 1989, p. 23.
%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 4th ed. (1975). Oxford, England: Oxford University Press. See 22.10, "The number of prime factors of n".
%D József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Section VII.28, p. 257.
%H Robert G. Wilson v, <a href="/A077761/b077761.txt">Table of n, a(n) for n = 0..5000</a>
%H Christian Axler, <a href="http://math.colgate.edu/~integers/s52/s52.Abstract.html">New estimates for some functions defined over primes</a>, Integers, Vol. 18 (2018), Article #A52.
%H Chris Caldwell, The Prime Pages, <a href="https://t5k.org/infinity.shtml#punchline">There are infinitely many primes, but, how big of an infinity?</a>
%H Henri Cohen, <a href="http://www.math.u-bordeaux.fr/~cohen/hardylw.dvi">High precision computation of Hardy-Littlewood constants</a>, preprint, 1998. - From _N. J. A. Sloane_, Jan 26 2013
%H Pierre Dusart, <a href="https://doi.org/10.1007/s11139-016-9839-4">Explicit estimates of some functions over primes</a>, The Ramanujan Journal, Vol. 45 (2018), pp. 227-251.
%H Pierre Dusart, <a href="https://doi.org/10.37394/23206.2023.22.57">On the divergence of the sum of prime reciprocals</a>, WSEAS Transactions on Math. (2023) Vol.22, 508-513.
%H Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 203.
%H Philippe Flajolet and Ilan Vardi, <a href="http://algo.inria.fr/flajolet/Publications/landau.ps">Zeta function expansions of some classical constants</a>.
%H Pieter Moree, <a href="https://web.archive.org/web/20050320062638/http://web.inter.nl.net/hcc/J.Moree/linnumb.htm">Mathematical constants</a>.
%H Dimbinaina Ralaivaosaona and Faratiana Brice Razakarinoro, <a href="https://arxiv.org/abs/2001.05782">An explicit upper bound for Siegel zeros of imaginary quadratic fields</a>, arXiv:2001.05782 [math.NT], 2020.
%H Xavier Gourdon and Pascal Sebah, <a href="http://numbers.computation.free.fr/Constants/constants.html">Constants from number theory</a>
%H Torsten Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/harmonic-series">The Harmonic Numbers and Series</a>.
%H Jonathan Sondow and Kieren MacMillan, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.124.3.232">Primary pseudoperfect numbers, arithmetic progressions, and the Erdos-Moser equation</a>, Amer. Math. Monthly, Vol. 124, No. 3 (2017), pp. 232-240; also on <a href="http://arxiv.org/abs/1812.06566">arXiv preprint</a>, arXiv:math/1812.06566 [math.NT], 2018.
%H Mark B. Villarino, <a href="https://arxiv.org/abs/math/0504289">Mertens' proof of Mertens' Theorem</a>, arXiv:math/0504289 [math.HO], 2005.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MertensConstant.html">Mertens Constant</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeZetaFunction.html">Prime Zeta Function</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HarmonicSeriesofPrimes.html">Harmonic Series of Primes</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Meissel%E2%80%93Mertens_constant">Meissel-Mertens constant</a>.
%H Marek Wójtowicz, <a href="http://dx.doi.org/10.3792/pjaa.87.22">Another proof on the existence of Mertens's constant</a>, Proc. Japan Acad. Ser. A Math. Sci., Vol. 87, No. 2 (2011), pp. 22-23.
%F Equals A001620 - Sum_{n>=2} zeta_prime(n)/n where the zeta prime sequence is A085548, A085541, A085964, A085965, A085966 etc. [Sebah and Gourdon] - _R. J. Mathar_, Apr 29 2006
%F Equals gamma + Sum_{p prime} (log(1-1/p) + 1/p), where gamma is Euler's constant (A001620). - _Amiram Eldar_, Dec 25 2021
%e 0.26149721284764278375542683860869585905156664826119920619206421392...
%t $MaxExtraPrecision = 400; RealDigits[ N[EulerGamma + NSum[(MoebiusMu[m]/m)*Log[N[Zeta[m], 120]], {m, 2, 1000}, Method -> "EulerMaclaurin", AccuracyGoal -> 120, NSumTerms -> 1000, PrecisionGoal -> 120, WorkingPrecision -> 120] , 120]][[1, 1 ;; 105]]
%t (* or, from version 7 up: *) digits = 105; M = EulerGamma - NSum[ PrimeZetaP[n] / n, {n, 2, Infinity}, WorkingPrecision -> digits+10, NSumTerms -> 3*digits]; RealDigits[M, 10, digits] // First (* _Jean-François Alcover_, Mar 16 2011, updated Sep 01 2015 *)
%Y Cf. A001620.
%K cons,nonn
%O 0,1
%A _T. D. Noe_, Nov 14 2002
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