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A343612
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Decimal expansion of P_{3,2}(2) = Sum 1/p^2 over primes == 2 (mod 3).
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10
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3, 0, 7, 9, 2, 0, 7, 5, 8, 6, 0, 7, 7, 3, 6, 4, 3, 6, 8, 4, 2, 5, 0, 5, 0, 7, 5, 9, 4, 0, 9, 9, 8, 7, 2, 6, 5, 8, 1, 0, 3, 2, 6, 6, 5, 4, 7, 5, 5, 1, 4, 4, 8, 0, 0, 5, 2, 0, 1, 9, 2, 5, 2, 9, 9, 3, 7, 8, 5, 5, 4, 9, 0, 1, 1, 2, 5, 6, 3, 3, 4, 3, 4, 8, 9, 0, 2, 2, 5, 9, 2, 4, 9, 3, 7, 8, 6, 8, 8, 9, 5, 1, 9, 5, 0
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OFFSET
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0,1
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COMMENTS
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The prime zeta modulo function P_{m,r}(s) = Sum_{primes p == r (mod m)} 1/p^s generalizes the prime zeta function P(s) = Sum_{primes p} 1/p^s.
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LINKS
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FORMULA
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EXAMPLE
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0.30792075860773643684250507594099872658103266547551448005201925299378554901...
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MATHEMATICA
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digits = 105; nmax0 = 20; dnmax = 5;
Clear[PrimeZeta31];
PrimeZeta31[s_, nmax_] := PrimeZeta31[s, nmax] = 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, nmax}] // N[#, digits+5]&;
PrimeZeta31[2, nmax = nmax0];
PrimeZeta31[2, nmax += dnmax];
While[Abs[PrimeZeta31[2, nmax] - PrimeZeta31[2, nmax-dnmax]] > 10^-(digits+5), Print["nmax = ", nmax]; nmax += dnmax];
PrimeZeta32[2] = PrimeZetaP[2] - 1/3^2 - PrimeZeta31[2, nmax];
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PROG
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(PARI)
s=0; forprimestep(p=2, 1e8, 3, s+=1./p^2); s \\ For illustration: using primes up to 10^N gives about 2N+2 (= 18 for N=8) correct digits.
PrimeZeta32(s)={sumeulerrat(1/p^s)-1/3^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))}
A343612_upto(N=100)={localprec(N+5); digits(PrimeZeta32(2)\.1^N)}
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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