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A343625
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Decimal expansion of the Prime Zeta modulo function P_{3,1}(5) = Sum 1/p^5 over primes p == 1 (mod 3).
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3
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0, 0, 0, 0, 6, 2, 6, 5, 5, 4, 2, 7, 4, 7, 1, 7, 5, 5, 5, 0, 6, 0, 0, 2, 5, 6, 9, 1, 9, 1, 0, 2, 4, 0, 8, 8, 4, 4, 6, 4, 7, 5, 7, 2, 0, 6, 7, 2, 6, 2, 0, 8, 2, 4, 1, 0, 6, 9, 5, 1, 6, 1, 4, 3, 6, 3, 6, 9, 7, 5, 1, 8, 8, 8, 4, 1, 3, 4, 3, 0, 7, 9, 7, 0, 3, 6, 1, 4, 6, 9, 3, 7, 9, 9, 5, 1, 9, 7, 3, 3
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OFFSET
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0,5
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COMMENTS
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The Prime Zeta modulo function at 5 for primes of the form 3k+1 is Sum_{primes in A002476} 1/p^5 = 1/7^5 + 1/13^5 + 1/19^5 + 1/31^5 + ...
The complementary Sum_{primes in A003627} 1/p^5 is given by P_{3,2}(5) = A085965 - 1/3^5 - (this value here) = 0.03157713571900394195603378... = A343615.
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LINKS
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EXAMPLE
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P_{3,1}(5) = 6.2655427471755506002569191024088446475720672620824106951614...*10^-5
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MATHEMATICA
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With[{s=5}, 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 *)
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PROG
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(PARI) s=0; forprimestep(p=1, 1e8, 3, s+=1./p^5); s \\ Naïve, for illustration: primes up to 10^N give 4N+2 (= 34 for N=8) correct digits.
(PARI) A343607_upto(N=100)={localprec(N+5); digits((PrimeZeta31(5)+1)\.1^N)[^1]} \\ cf. A175644 for PrimeZeta31
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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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