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

Author

R. J. Mathar, Aug 01 2010

Keywords

• cons
• nonn

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.

Examples

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

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.

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)).

Programs

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 *)

PARI

my(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. - M. F. Hasler, Apr 23 2021

PARI

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]} \\ M. F. Hasler, Apr 23 2021

Extensions

More digits from Vaclav Kotesovec, Jun 27 2020