A343612 - OEIS

A343612

Decimal expansion of P_{3,2}(2) = Sum 1/p^2 over primes == 2 (mod 3).

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

0,1

COMMENTS

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.

LINKS

Jean-François Alcover, Table of n, a(n) for n = 0..1005

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.

FORMULA

P_{3,2}(2) = P(2) - 1/3^2 - P_{3,1}(2) = A085548 - A000012 - A175644.

EXAMPLE

0.30792075860773643684250507594099872658103266547551448005201925299378554901...

MATHEMATICA

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];

RealDigits[PrimeZeta32[2], 10, digits][[1]] (* Jean-François Alcover, May 06 2021, after M. F. Hasler's PARI code *)

PROG

(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)}

CROSSREFS

Cf. A003627 (primes 3k-1), A085548 (PrimeZeta(2)), A021031 (1/27).

Cf. A175644 (same for primes 3k+1), A086032 (for primes 4k+1), A085991 (for primes 4k+3), A343613 - A343619 (P_{3,2}(s): same with 1/p^s, s = 3, ..., 9).

Sequence in context: A019970 A342698 A239022 * A363502 A198488 A296364

Adjacent sequences: A343609 A343610 A343611 * A343613 A343614 A343615

KEYWORD

nonn,cons

AUTHOR

M. F. Hasler, Apr 22 2021

STATUS

approved