

A077761


Decimal expansion of Mertens' 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 ith prime number.


9



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

0,1


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]


REFERENCES

S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2004, pp. 9498
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, A Foundation For Computer Science, AddisonWesley, Reading, MA, 1989, p. 23.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, 1996, Section VII.28, p. 257.


LINKS

Table of n, a(n) for n=0..104.
Christian Axler, New estimates for some functions defined over primes, Integers (2018) 18, Article #A52.
Chris Caldwell, The Prime Pages, There are infinitely many primes, but, how big of an infinity?
H. Cohen, High precision computation of HardyLittlewood constants, preprint, 1998.  From N. J. A. Sloane, Jan 26 2013
Pierre Dusart, Explicit estimates of some functions over primes, The Ramanujan Journal, 2016.
Ph. Flajolet and I. Vardi, Zeta function expansions of some classical constants
Pieter Moree, Mathematical constants
P. Sebah and X. Gourdon, Constants from number theory
Torsten Sillke, The Harmonic Numbers and Series.
J. Sondow and K. MacMillan, Primary pseudoperfect numbers, arithmetic progressions, and the ErdosMoser equation, Amer. Math. Monthly, 124 (2017) 232240; arXiv:math/1812.06566 [math.NT], 2018.
M. B. Villarino, Mertens' proof of Mertens' Theorem
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


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


EXAMPLE

0.26149721284764278375542683860869585905156664826119920619206421392...


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 (* JeanFrançois Alcover, Mar 16 2011, updated Sep 01 2015 *)


CROSSREFS

Sequence in context: A192043 A154584 A129677 * A220406 A220794 A220959
Adjacent sequences: A077758 A077759 A077760 * A077762 A077763 A077764


KEYWORD

cons,nonn,changed


AUTHOR

T. D. Noe, Nov 14 2002


STATUS

approved



