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A077761
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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.
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69
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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
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OFFSET
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0,1
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 203.
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FORMULA
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Equals gamma + Sum_{p prime} (log(1-1/p) + 1/p), where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 25 2021
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EXAMPLE
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0.26149721284764278375542683860869585905156664826119920619206421392...
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MATHEMATICA
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$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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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