OFFSET
0,1
COMMENTS
It is not known if this series actually converges.
REFERENCES
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, pp. 94-98.
LINKS
Richard J. Mathar, Table of Dirichlet L-series..., arXiv:1008.2547 [math.NT], 2010-2015, value of S(m=3,r=2,s=1).
Eric Weisstein's World of Mathematics, Prime Sums.
FORMULA
EXAMPLE
0.64194483853319570866139263972173431667541104401588965490817...
MATHEMATICA
S[m_, n_, s_] := (t = 1; sums = 0; difs = 1; While[Abs[difs] > 10^(-digits - 5) || difs == 0, difs = (MoebiusMu[t]/t) * Log[If[s*t == 1, DirichletL[m, n, s*t], Sum[Zeta[s*t, j/m]*DirichletCharacter[m, n, j]^t, {j, 1, m}]/m^(s*t)]]; sums = sums + difs; t++]; sums); $MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[-S[3, 2, 1], digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 22 2021 *)
CROSSREFS
KEYWORD
AUTHOR
Eric W. Weisstein, Jul 13 2003
EXTENSIONS
More digits from R. J. Mathar, Jul 28 2010
Sign typo in definition corrected by R. J. Mathar, Aug 01 2010
STATUS
approved