A175644 Decimal expansion of the sum 1/p^2 over primes p == 1 (mod 3).

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

Offset

0,2

Comments

The prime zeta modulo function at 2 for primes of the form 3k+1, which is P_{3,2}(2) = Sum_{p in A002476} 1/p^2 = 1/7^2 + 1/13^2 + 1/19^2 + 1/31^2 + ...

The complementary Sum_{p in A003627} 1/p^2 is given by P_{3,2}(2) = A085548 - 1/3^2 - (this value here) = 0.307920758607736436842505075940... = A343612.

Links

Jean-François Alcover, Table of n, a(n) for n = 0..1003

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.

OEIS index to entries related to the (prime) zeta function.

Example

P_{3,1}(2) = 0.03321555032221795055292717778013809648108756665...

Mathematica

With[{s=2}, Do[Print[N[1/2 * Sum[(MoebiusMu[2*n + 1]/(2*n + 1)) * Log[(Zeta[s + 2*n*s]*(Zeta[s + 2*n*s, 1/6] - Zeta[s + 2*n*s, 5/6])) / ((1 + 2^(s + 2*n*s))*(1 + 3^(s + 2*n*s)) * Zeta[2*(1 + 2*n)*s])], {n, 0, m}], 120]], {m, 100, 500, 100}]] (* Vaclav Kotesovec, Jan 13 2021 *)
digits = 1003;
m = 100; (* initial value of n beyond which summand is considered negligible *)
dm = 100; (* increment of m *)
P[s_, m_] (* "P" short name for "PrimeZeta31" *):= P[s, m] = Sum[Module[{t}, t = s + 2 n*s; MoebiusMu[2n + 1]* ((1/(4n + 2)) (-Log[1 + 2^t] - Log[1 + 3^t] + Log[Zeta[t]] - Log[Zeta[2t]] + Log[Zeta[t, 1/6] - Zeta[t, 5/6]]))], {n, 0, m}] // N[#, digits+10]&;
P[2, m]; P[2, m += dm];
While[ RealDigits[P[2,    m]][[1]][[1;; digits]] !=
       RealDigits[P[2, m-dm]][[1]][[1;; digits]], Print["m = ", m]; m+=dm];
Join[{0}, RealDigits[P[2, m]][[1]][[1;; digits]]] (* Jean-François Alcover, Jun 24 2022, after Vaclav Kotesovec *)

Prog

(PARI) From M. F. Hasler, Apr 23 2021: (Start)
s=0; forprimestep(p=1, 1e8, 3, s+=1./p^2); s \\ For illustration: primes up to 10^N give only about 2N+2 (= 18 for N=8) correct digits.
PrimeZeta31(s)=suminf(n=0, my(t=s+2*n*s); moebius(2*n+1)*log((zeta(t)*(zetahurwitz(t, 1/6)-zetahurwitz(t, 5/6)))/((1+2^t)*(1+3^t)*zeta(2*t)))/(4*n+2)) \\ Inspired from Kotesovec's Mmca code
A175644_upto(N=100)={localprec(N+5); digits((PrimeZeta31(2)+1)\.1^N)[^1]} \\ (End)

Crossrefs

Cf. A086032 (P_{4,1}(2): same for p==1 (mod 4)), A175645 (P_{3,1}(3): same for 1/p^3), A343612 (P_{3,2}(2): same for p==2 (mod 3)), A085548 (PrimeZeta(2)).

Sequence in context: A038766 A080993 A140259 * A102905 A020862 A131589

Adjacent sequences: A175641 A175642 A175643 * A175645 A175646 A175647

Keyword

cons,nonn

Author

R. J. Mathar, Aug 01 2010

Extensions

More digits from Vaclav Kotesovec, Jun 27 2020

Status

approved