A086036 -id:A086036 - cp's OEIS frontend

A085966 Decimal expansion of the prime zeta function at 6.

0, 1, 7, 0, 7, 0, 0, 8, 6, 8, 5, 0, 6, 3, 6, 5, 1, 2, 9, 5, 4, 1, 3, 3, 6, 7, 3, 2, 6, 6, 0, 5, 9, 3, 9, 9, 2, 0, 9, 5, 8, 5, 9, 4, 1, 8, 7, 4, 5, 4, 4, 2, 4, 4, 7, 3, 3, 1, 6, 3, 3, 6, 8, 8, 3, 6, 9, 6, 9, 7, 3, 6, 7, 4, 7, 1, 7, 2, 4, 3, 6, 6, 7, 1, 8, 6, 0, 3, 5, 0, 0, 7, 8, 1, 8, 0, 6, 2, 3, 0, 2, 8, 8, 2, 3

Offset: 0

Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003

Mathar's Table 1 (cited below) lists expansions of the prime zeta function at integers s in 10..39. - Jason Kimberley, Jan 07 2017

			0.0170700868506365129541...

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
J. W. L. Glaisher, On the Sums of Inverse Powers of the Prime Numbers, Quart. J. Math. 25, 347-362, 1891.

Jason Kimberley, Table of n, a(n) for n = 0..1802
Henri Cohen, High Precision Computation of Hardy-Littlewood Constants, Preprint, 1998.
Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
X. Gourdon and P. Sebah, Some Constants from Number theory
R. J. Mathar, Series of reciprocal powers of k-almost primes, arXiv:0803.0900 [math.NT], 2008-2009. Table 1.
Eric Weisstein's World of Mathematics, Prime Zeta Function

Decimal expansion of the prime zeta function: A085548 (at 2), A085541 (at 3), A085964 (at 4), A085965 (at 5), this sequence (at 6), A085967 (at 7) to A085969 (at 9).

Cf. A013664, A030516.

R := RealField(106);
PrimeZeta := func;
[0]cat Reverse(IntegerToSequence(Floor(PrimeZeta(6,57)*10^105)));
// Jason Kimberley, Dec 30 2016

s[n_] := s[n] = Sum[ MoebiusMu[k]*Log[Zeta[6*k]]/k, {k, 1, n}] // RealDigits[#, 10, 104]& // First // Prepend[#, 0]&; s[100]; s[n = 200]; While[s[n] != s[n - 100], n = n + 100]; s[n] (* Jean-François Alcover, Feb 14 2013 *)
RealDigits[ PrimeZetaP[ 6], 10, 111][[1]] (* Robert G. Wilson v, Sep 03 2014 *)

sumeulerrat(1/p,6) \\ Hugo Pfoertner, Feb 03 2020

P(6) = Sum_{p prime} 1/p^6 = Sum_{n>=1} mobius(n)*log(zeta(6*n))/n
Equals 1/2^6 + A085995 + A086036. - R. J. Mathar, Jul 14 2012
Equals Sum_{k>=1} 1/A030516(k). - Amiram Eldar, Jul 27 2020

A085995 Decimal expansion of the prime zeta modulo function at 6 for primes of the form 4k+3.

Original entry on oeis.org

0, 0, 1, 3, 8, 0, 8, 3, 5, 8, 8, 6, 9, 7, 1, 7, 3, 9, 1, 6, 3, 0, 3, 1, 8, 5, 4, 1, 2, 8, 0, 1, 5, 8, 2, 2, 6, 1, 0, 6, 0, 1, 3, 9, 6, 3, 2, 7, 5, 6, 5, 4, 2, 9, 6, 8, 0, 2, 6, 4, 8, 0, 2, 5, 7, 8, 5, 3, 0, 7, 5, 2, 2, 2, 7, 0, 7, 4, 6, 9, 1, 3, 4, 7, 9, 1, 5, 6, 0, 4, 2, 5, 1, 7, 1, 0, 1, 6, 6, 0, 1, 6, 8, 7, 8

Offset: 0

Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003

			0.0013808358869717391630318541280158226106013963275654296802648025785307522...

P. Flajolet and I. Vardi, Zeta Function Expansions of Classical Constants, Unpublished manuscript. 1996.
X. Gourdon and P. Sebah, Some Constants from Number theory.
R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015, value P(m=4, n=3, s=6), page 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A002145 (primes 4k+3), A001014 (n^6), A085966 (PrimeZeta(6)).

Cf. A085991 - A085998 (Zeta_R(2..9): same for 1/p^2, ..., 1/p^9), A086036 (same for primes 4k+1), A343626 (for primes 3k+1), A343616 (for primes 3k+2).

b[x_] = (1 - 2^(-x))*(Zeta[x]/DirichletBeta[x]); $MaxExtraPrecision = 250; m = 40; Join[{0, 0}, RealDigits[(1/2)*NSum[MoebiusMu[2n + 1]* Log[b[(2n + 1)*6]]/(2n + 1), {n, 0, m}, AccuracyGoal -> 120, NSumTerms -> m, PrecisionGoal -> 120, WorkingPrecision -> 120] ][[1]]][[1 ;; 105]] (* Jean-François Alcover, Jun 22 2011, updated Mar 14 2018 *)

A085995_upto(N=100)={localprec(N+3); digits((PrimeZeta43(6)+1)\.1^N)[^1]} \\ see A085991 for the PrimeZeta43 function. - M. F. Hasler, Apr 25 2021

Zeta_R(6) = Sum_{p in A002145} 1/p^6 where A002145 = {primes p == 3 (mod 4)},
  = (1/2)*Sum_{n >= 0} möbius(2*n+1)*log(b((2*n+1)*6))/(2*n+1),
  where b(x) = (1-2^(-x))*zeta(x)/L(x) and L(x) is the Dirichlet Beta function.

Edited by M. F. Hasler, Apr 25 2021

A343626 Decimal expansion of the Prime Zeta modulo function P_{3,1}(6) = Sum 1/p^6 over primes p == 1 (mod 3).

Original entry on oeis.org

0, 0, 0, 0, 0, 8, 7, 3, 0, 0, 1, 1, 0, 2, 3, 1, 9, 8, 1, 6, 7, 0, 1, 2, 0, 4, 2, 7, 7, 9, 1, 4, 5, 2, 3, 1, 9, 4, 9, 5, 6, 1, 0, 7, 9, 7, 6, 4, 5, 3, 9, 1, 8, 3, 6, 9, 8, 9, 7, 1, 7, 7, 1, 3, 8, 1, 3, 6, 2, 9, 8, 3, 2, 9, 4, 5, 3, 8, 7, 6, 4, 9, 6, 9, 9, 3, 6, 1, 8, 5, 8, 6, 2, 3, 2, 9, 3, 3, 4, 5

Offset: 0

M. F. Hasler, Apr 23 2021

The Prime Zeta modulo function at 6 for primes of the form 3k+1 is Sum_{primes in A002476} 1/p^6 = 1/7^6 + 1/13^6 + 1/19^6 + 1/31^6 + ...

The complementary Sum_{primes in A003627} 1/p^6 is given by P_{3,2}(6) = A085966 - 1/3^6 - (this value here) = 0.015689614727130461563527666... = A343606.

			P_{3,1}(6) = 8.7300110231981670120427791452319495610797645391837...*10^-8

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.
OEIS index to entries related to the (prime) zeta function.

Cf. A175645, A343624 - A343629 (P_{3,1}(3..9): same for 1/p^n, n=3..9), A343606 (P_{3,2}(6): same for p==2 (mod 3)), A086036 (P_{4,1}(6): same for p==1 (mod 4)).

Cf. A085966 (PrimeZeta(6)), A002476 (primes of the form 3k+1).

With[{s=6}, 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 *)

s=0; forprimestep(p=1, 1e8, 3, s+=1./p^6); s \\ For illustration: primes up to 10^N give 5N+2 (= 42 for N=8) correct digits.

A343626_upto(N=100)={localprec(N+5);digits((PrimeZeta31(6)+1)\.1^N)[^1]} \\ cf. A175644 for PrimeZeta31

A343616 Decimal expansion of P_{3,2}(6) = Sum 1/p^6 over primes == 2 (mod 3).

Original entry on oeis.org

0, 1, 5, 6, 8, 9, 6, 1, 4, 7, 2, 7, 1, 3, 0, 4, 6, 1, 5, 6, 3, 5, 2, 7, 6, 6, 6, 1, 5, 2, 2, 0, 9, 0, 9, 1, 8, 1, 4, 2, 0, 8, 6, 7, 5, 5, 5, 3, 0, 7, 7, 7, 6, 3, 3, 6, 6, 1, 5, 3, 1, 8, 8, 6, 7, 6, 4, 5, 7, 2, 3, 3, 5, 6, 2, 3, 7, 3, 0, 4, 0, 7, 0, 0, 5, 5, 2, 4, 2, 2, 1, 0, 3, 3, 6, 8, 4, 3, 5, 2

Offset: 0

M. F. Hasler, Apr 25 2021

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.

			0.015689614727130461563527666152209091814208675553077763366153188676457...

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, value P(m=3, n=2, s=6), p. 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A003627 (primes 3k-1), A001014 (n^6), A085966 (PrimeZeta(6)), A021733 (1/3^6).
Cf. A343612 - A343619 (P_{3,2}(s): analog for 1/p^s, s = 2 .. 9).
Cf. A343626 (for primes 3k+1), A086036 (for primes 4k+1), A085995 (for primes 4k+3).

A343616_upto(N=100)={localprec(N+5); digits((PrimeZeta32(6)+1)\.1^N)[^1]} \\ see A343612 for the function PrimeZeta32

P_{3,2}(6) = Sum_{p in A003627} 1/p^6 = P(6) - 1/3^6 - P_{3,1}(6).

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A085966 Decimal expansion of the prime zeta function at 6.

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A085995 Decimal expansion of the prime zeta modulo function at 6 for primes of the form 4k+3.

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A343626 Decimal expansion of the Prime Zeta modulo function P_{3,1}(6) = Sum 1/p^6 over primes p == 1 (mod 3).

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A343616 Decimal expansion of P_{3,2}(6) = Sum 1/p^6 over primes == 2 (mod 3).

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