A345327 Decimal expansion of a constant Y2 related to the asymptotics of A000203.
5, 0, 7, 3, 3, 8, 8, 8, 2, 5, 8, 3, 0, 8, 4, 3, 7, 8, 1, 0, 0, 4, 9, 7, 8, 7, 6, 5, 1, 5, 9, 5, 2, 6, 7, 7, 3, 8, 9, 0, 1, 9, 6, 3, 4, 8, 2, 8, 1, 6, 4, 4, 8, 0, 8, 0, 4, 9, 7, 4, 5, 8, 7, 7, 2, 4, 5, 0, 6, 9, 4, 6, 1, 7, 3, 0, 2, 8, 6, 5, 1, 6, 3, 0, 0, 5, 6, 8, 8, 3, 9, 1, 7, 6, 3, 0, 2, 4, 6, 5, 9, 6, 0, 5, 8, 0
Offset: 0
Examples
0.5073388825830843781004978765159526773890196348281644808049...
Links
- Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 51 (constant Y2).
- V. Sita Ramaiah and D. Suryanarayana, Sums of reciprocals of some multiplicative functions, Mathematical Journal of Okayama University, Vol. 21, No. 2 (1979), pp. 161-162, formula (4.1)-(4.4).
- László Tóth, Alternating sums concerning multiplicative arithmetic functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1 (section 4.3, p. 18).
Programs
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Mathematica
$MaxExtraPrecision = 1000; Do[ratfun = (p - 1)^2 * Sum[j/(p^j - 1)/(p^(j + 1) - 1), {j, 1, m}]/(p*(1 - (p - 1)^2/p * Sum[1/(p^j - 1)/(p^(j + 1) - 1), {j, 1, m}])); zetas = 0; ratab = Table[konfun = Together[ratfun + c/(p^power - 1)]; coefs = CoefficientList[Numerator[konfun], p]; sol = Solve[Last[coefs] == 0, c][[1]]; zetas = zetas + c*Zeta'[power]/Zeta[power] /. sol; ratfun = konfun /. sol, {power, 2, 40}]; Print[N[Sum[Log[p]*ratfun /. p -> Prime[k], {k, 1, m}] + zetas, 110]], {m, 10, 250, 10}]