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 A343624 Decimal expansion of the Prime Zeta modulo function P_{3,1}(4) = Sum 1/p^4 over primes p == 1 (mod 3). 7
 0, 0, 0, 4, 6, 1, 3, 1, 5, 0, 5, 5, 3, 4, 3, 3, 8, 6, 9, 4, 0, 1, 7, 4, 5, 3, 0, 3, 3, 3, 4, 0, 9, 4, 5, 4, 3, 3, 9, 9, 3, 9, 0, 1, 8, 3, 5, 3, 8, 1, 6, 8, 7, 0, 3, 6, 7, 9, 6, 6, 8, 3, 7, 5, 9, 6, 2, 4, 8, 9, 7, 8, 8, 5, 3, 2, 7, 9, 5, 2, 8, 8, 5, 0, 0, 2, 1, 9, 0, 0, 8, 5, 6, 6, 6, 8, 3, 6, 9, 7 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS The Prime Zeta modulo function at 4 for primes of the form 3k+1 is Sum_{primes in A002476} 1/p^4 = 1/7^4 + 1/13^4 + 1/19^4 + 1/31^4 + ... The complementary Sum_{primes in A003627} 1/p^4 is given by P_{3,2}(4) = A085964 - 1/3^4 - (this value here) = 0.064186145696557778990099... = A343614. LINKS R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, p.21 EXAMPLE P_{3,1}(4) = 0.000461315055343386940174530333409454339939018353816870... MATHEMATICA With[{s=4}, 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}]] (* adopted from Vaclav Kotesovec's code in A175645 *) PROG (PARI) s=0; forprimestep(p=1, 1e8, 3, s+=1./p^4); s \\ Naïve, for illustration: primes up to 10^N give about 3N+2 (= 26 for N=8) correct digits. (PARI) A343606_upto(N=100)={localprec(N+5); digits((PrimeZeta31(4)+1)\.1^N)[^1]} \\ cf. A175644 for PrimeZeta31 CROSSREFS Cf. A175645, A343625 - A343629 (P_{3,1}(3..9): same for 1/p^s, s=3, 5,..., 9). Cf. A343614 (P_{3,2}(4): same for p==2 (mod 3)), A086034 (P_{4,1}(4): same for p==1 (mod 4)), A085964 (PrimeZeta(4)). Sequence in context: A154478 A255695 A246489 * A309445 A051261 A247621 Adjacent sequences:  A343617 A343618 A343619 * A343625 A343626 A343627 KEYWORD cons,nonn AUTHOR M. F. Hasler, Apr 23 2021 STATUS approved

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Last modified June 24 17:34 EDT 2021. Contains 345418 sequences. (Running on oeis4.)