OFFSET
1,3
COMMENTS
Or, decimal expansion of constant B2 from the summatory function of the restricted divisor function.
The constant A in the asymptotic formula Sum_{prime p <= n} 1/(p-1) = log(log(n)) + A + O(1/log(n)) (Jakimczuk, 2017). - Amiram Eldar, Mar 18 2024
REFERENCES
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, Vol. 94, Cambridge University Press, 2003, pp. 94-98.
József Sándor and Borislav Crstici, Handbook of Number Theory II, Kluwer Academic Publishers, 2004, p. 155, Chapter V, 1) b).
LINKS
Rafael Jakimczuk, On Sums of Powers of the p-adic Valuation of n!, J. Int. Seq. (2017) Vol. 20, Art. 17.5.6.
Dimbinaina Ralaivaosaona and Faratiana Brice Razakarinoro, An explicit upper bound for Siegel zeros of imaginary quadratic fields, arXiv:2001.05782 [math.NT], 2020.
Dimbinaina Ralaivaosaona and Faratiana Brice Razakarinoro, An explicit bound for Siegel zeros and the torsion of elliptic curves with complex multiplication, J. Num. Theory (2026) Vol. 281, 795-829. See p. 799.
Terence Tao and Joni Teräväinen, Quantitative correlations and some problems on prime factors of consecutive integers, arXiv:2512.01739 [math.NT], 2025. See p. 2.
Eric Weisstein's World of Mathematics, Mertens Constant.
Eric Weisstein's World of Mathematics, Prime Factor.
FORMULA
Equals gamma + Sum_{p prime} (log(1-1/p) + 1/(p-1)), where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 25 2021
From Amiram Eldar, Mar 18 2024: (Start)
Equals gamma + Sum_{k>=2} phi(k) * log(zeta(k)) / k, where phi = A000010.
Equals gamma - Sum_{p prime} 1/(p-1)^2 + Sum_{k>=2} J_2(k) * log(zeta(k)) / k, where J_2 = A007434.
Both formulas are from Jakimczuk (2017). (End)
EXAMPLE
1.03465388189743791161979429846463825467030798434438525450307...
MATHEMATICA
digits = 102; Mp = EulerGamma - NSum[PrimeZetaP[n]/n - PrimeZetaP[n], {n, 2, Infinity}, WorkingPrecision -> digits + 10, NSumTerms -> 3*digits]; RealDigits[Mp, 10, digits] // First (* Jean-François Alcover, Sep 02 2015 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Eric W. Weisstein, Sep 25 2003
STATUS
approved