A358789 Decimal expansion of Sum_{p prime, p>=3} (-1)^((p-1)/2)*log(p)/p, negated.

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

Offset

0,1

Comments

Sum_{p prime} log(p)/p is divergent.

Links

Table of n, a(n) for n=0..114.

Formula

Limit_{N->oo} ((Sum_{p<=N prime == 3 (mod 4)} log(p)/p) - (Sum_{p<=N prime == 1 (mod 4)} log(p)/p)).

Example

-0.54568127279512790148953238338...

Mathematica

alfa[s_]:= 1/(1 + 1/2^s) * DirichletBeta[s] * Zeta[s] / Zeta[2*s]; beta[s_]:= (1 - 1/2^s) * Zeta[s] / DirichletBeta[s]; Do[Print[N[-1/2*Sum[MoebiusMu[2*n + 1]/(2*n + 1) * Limit[D[Log[alfa[(2*n + 1)*s]/beta[(2*n + 1)*s]], s], s -> 1], {n, 0, m}], 120]], {m, 20, 200, 20}] (* Vaclav Kotesovec, Jan 25 2023 *)

Crossrefs

Cf. A086239, A091812, A136271, A354295.

Sequence in context: A021651 A200293 A211006 * A069214 A119807 A254181

Adjacent sequences: A358786 A358787 A358788 * A358790 A358791 A358792

Keyword

nonn,cons

Author

Artur Jasinski, Jan 03 2023

Status

approved