A086241 Decimal expansion of value to which Sum_{k>=2} d(k)/prime(k) appears to converge, where d(k)=-1 if p mod 3 = 1, d(k)=+1 if p mod 3 = 2 and d(k)=0 if p mod 3 = 0.

6, 4, 1, 9, 4, 4, 8, 3, 8, 5, 3, 3, 1, 9, 5, 7, 0, 8, 6, 6, 1, 3, 9, 2, 6, 3, 9, 7, 2, 1, 7, 3, 4, 3, 1, 6, 6, 7, 5, 4, 1, 1, 0, 4, 4, 0, 1, 5, 8, 8, 9, 6, 5, 4, 9, 0, 8, 1, 7, 0, 8, 4, 5, 1, 3, 1, 7, 3, 3, 2, 8, 2, 6, 9, 0, 7, 3, 7, 2, 3, 3, 5, 9, 8, 1, 7, 4, 1, 5, 9, 9, 4, 5, 6, 0, 6, 5, 7

Offset: 0

Author

Eric W. Weisstein, Jul 13 2003

Keywords

• nonn
• cons
• hard

Comments

It is not known if this series actually converges.

Examples

			0.64194483853319570866139263972173431667541104401588965490817...
		

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.

Crossrefs

Cf. A161529, A368644.

Programs

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 *)

Formula

Equals A161529 + A368644. - Amiram Eldar, Jan 02 2024

Extensions

More digits from R. J. Mathar, Jul 28 2010

Sign typo in definition corrected by R. J. Mathar, Aug 01 2010