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%I #28 Jun 24 2022 11:10:41
%S 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,
%T 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,
%U 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
%N Decimal expansion of the sum 1/p^2 over primes p == 1 (mod 3).
%C 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 + ...
%C 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.
%H Jean-François Alcover, <a href="/A175644/b175644.txt">Table of n, a(n) for n = 0..1003</a>
%H R. J. Mathar, <a href="http://arxiv.org/abs/1008.2547">Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli</a>, arXiv:1008.2547 [math.NT], 2010-2015.
%H <a href="/index/Z#zeta_function">OEIS index to entries related to the (prime) zeta function</a>.
%e P_{3,1}(2) = 0.03321555032221795055292717778013809648108756665...
%t 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 *)
%t digits = 1003;
%t m = 100; (* initial value of n beyond which summand is considered negligible *)
%t dm = 100; (* increment of m *)
%t 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]&;
%t P[2, m]; P[2, m += dm];
%t While[ RealDigits[P[2, m]][[1]][[1;;digits]] !=
%t RealDigits[P[2, m-dm]][[1]][[1;;digits]], Print["m = ", m]; m+=dm];
%t Join[{0}, RealDigits[P[2, m]][[1]][[1;;digits]]] (* _Jean-François Alcover_, Jun 24 2022, after _Vaclav Kotesovec_ *)
%o (PARI) From _M. F. Hasler_, Apr 23 2021: (Start)
%o 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.
%o 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
%o A175644_upto(N=100)={localprec(N+5);digits((PrimeZeta31(2)+1)\.1^N)[^1]} \\ (End)
%Y 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)).
%K cons,nonn
%O 0,2
%A _R. J. Mathar_, Aug 01 2010
%E More digits from _Vaclav Kotesovec_, Jun 27 2020
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