A077761 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.

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, 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, 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

Offset: 0

Author

T. D. Noe, Nov 14 2002

Keywords

• cons
• nonn

Comments

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]

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

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

Examples

			0.26149721284764278375542683860869585905156664826119920619206421392...
		

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2004, pp. 94-98
• Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, A Foundation For Computer Science, Addison-Wesley, Reading, MA, 1989, p. 23.
• 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".
• József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Section VII.28, p. 257.

Links

• Eduard Baumann, Table of n, a(n) for n = 0..9999 (first 5001 digits from Robert G. Wilson v), Dec 03 2024.
• Christian Axler, New estimates for some functions defined over primes, Integers, Vol. 18 (2018), Article #A52.
• Chris Caldwell, The Prime Pages, There are infinitely many primes, but, how big of an infinity?
• Henri Cohen, High precision computation of Hardy-Littlewood constants, preprint, 1998. - From _N. J. A. Sloane_, Jan 26 2013
• Pierre Dusart, Explicit estimates of some functions over primes, The Ramanujan Journal, Vol. 45 (2018), pp. 227-251.
• Pierre Dusart, On the divergence of the sum of prime reciprocals, WSEAS Transactions on Math. (2023) Vol.22, 508-513.
• Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 203.
• Philippe Flajolet and Ilan Vardi, Zeta function expansions of some classical constants.
• Tengiz O. Gogoberidze, Baker's dozen digits of two sums involving reciprocal products of an integer and its greatest prime factor, arXiv:2407.12047 [math.GM], 2024. See p. 3.
• Peter Lindqvist and Jaak Peetre, On the remainder in a series of Mertens, Expos. Math. 15 (1997) 467-478.
• Peter Lindqvist and Jaak Peetre, On a number theoretic sum considered by Meissel : a historical observation, Nieuw Archief voor Wiskunde (Serie 4) 1997 Vol. 15 (3) pp. 175-179.
• Ernst Meissel, Bericht über die Provinzial-Gewerbe-Schule zu Iserlohn 1866. Title page [courtesy Archiv der Berlin-Brandenburgischen Akademie der Wissenschaften]
• Ernst Meissel, Bericht über die Provinzial-Gewerbe-Schule zu Iserlohn 1866. Notiz No. 55 [courtesy Archiv der Berlin-Brandenburgischen Akademie der Wissenschaften].
• Ernst Meissel, Ueber die Bestimmung der Primzahlenmenge innerhalb gegebener Grenzen, Math. Ann. 2 (4) (1870) 636-642, EuDML
• Pieter Moree, Mathematical constants.
• Rikard Olofsson, Properties of the Beurling generalized primes, Journal of Number Theory (Volume 131), Issue 1, January 2011, Pages 45-58 (p.51).
• Dimbinaina Ralaivaosaona and Faratiana Brice Razakarinoro, An explicit upper bound for Siegel zeros of imaginary quadratic fields, arXiv:2001.05782 [math.NT], 2020.
• Xavier Gourdon and Pascal Sebah, Constants from number theory
• Torsten Sillke, The Harmonic Numbers and Series.
• Jonathan Sondow and Kieren MacMillan, Primary pseudoperfect numbers, arithmetic progressions, and the Erdos-Moser equation, Amer. Math. Monthly, Vol. 124, No. 3 (2017), pp. 232-240; also on arXiv preprint, arXiv:math/1812.06566 [math.NT], 2018.
• Mark B. Villarino, Mertens' proof of Mertens' Theorem, arXiv:math/0504289 [math.HO], 2005.
• Eric Weisstein's World of Mathematics, Mertens Constant.
• Eric Weisstein's World of Mathematics, Prime Zeta Function.
• Eric Weisstein's World of Mathematics, Harmonic Series of Primes.
• Wikipedia, Meissel-Mertens constant.
• Marek Wójtowicz, Another proof on the existence of Mertens's constant, Proc. Japan Acad. Ser. A Math. Sci., Vol. 87, No. 2 (2011), pp. 22-23.

Crossrefs

Cf. A001620.

Programs

Mathematica

$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]]
(* 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 *)

Formula

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

Equals gamma + Sum_{p prime} (log(1-1/p) + 1/p), where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 25 2021

Equals lim_{k->oo} -k + Sum_{p prime} 1/(p*log(p)^(1/k)) conjectured by Meissel in 1866 and proven by Peter Lindqvist and Jaak Peetre in 1997 see links - Artur Jasinski, Mar 11 2025