A086035 -id:A086035 - cp's OEIS frontend

A085965 Decimal expansion of the prime zeta function at 5.

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

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 05 2017

			0.0357550174839242571328...

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..1702
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), this sequence (at 5), A085966 (at 6) to A085969 (at 9).

Cf. A013663, A050997, A242304.

R := RealField(106);
PrimeZeta := func;
[0] cat Reverse(IntegerToSequence(Floor(PrimeZeta(5,69)*10^105)));
// Jason Kimberley, Dec 30 2016

s[n_] := s[n] = Sum[ MoebiusMu[k]*Log[Zeta[5*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, from 1st formula *)
RealDigits[ PrimeZetaP[ 5], 10, 111][[1]] (* Robert G. Wilson v, Sep 03 2014 *)

sumeulerrat(1/p,5) \\ Hugo Pfoertner, Feb 03 2020

P(5) = Sum_{p prime} 1/p^5 = Sum_{n>=1} mobius(n)*log(zeta(5*n))/n.
Equals 1/2^5 + A085994 + A086035. - R. J. Mathar, Jul 14 2012
Equals Sum_{k>=1} 1/A050997(k). - Amiram Eldar, Jul 27 2020

A085994 Decimal expansion of the prime zeta modulo function at 5 for primes of the form 4k+3.

Original entry on oeis.org

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

Offset: 0

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

			0.004181543449702459614306334352814627154254543020852184353396741251345574...

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=5), page 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A085991 .. A085998 (Zeta_R(2..9)).

Cf. A086035 (analog for primes 4k+1), A085965 (PrimeZeta(5)), A002145 (primes 4k+3).

b[x_] = (1 - 2^(-x))*(Zeta[x]/DirichletBeta[x]); $MaxExtraPrecision = 200; m = 40; Join[{0, 0}, RealDigits[(1/2)*NSum[MoebiusMu[2n + 1]* Log[b[(2n + 1)*5]]/(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 *)

A085994_upto(N=100)={localprec(N+3); digits((PrimeZeta43(5)+1)\.1^N)[^1]} \\ see A085991 for the PrimeZeta43 function. - M. F. Hasler, Apr 25 2021

Zeta_R(5) = Sum_{primes r == 3 mod 4} 1/p^5
  = (1/2)*Sum_{n=0..inf} mobius(2*n+1)*log(b((2*n+1)*5))/(2*n+1),
  where b(x) = (1-2^(-x))*zeta(x)/L(x) and L(x) is the Dirichlet Beta function.

A343625 Decimal expansion of the Prime Zeta modulo function P_{3,1}(5) = Sum 1/p^5 over primes p == 1 (mod 3).

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Apr 23 2021

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.

			P_{3,1}(5) = 6.2655427471755506002569191024088446475720672620824106951614...*10^-5

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. A086035 (P_{4,1}(5): same for p==1 (mod 4)), A175645, A343624 - A343629 (P_{3,1}(3..9): same for 1/p^s, s=3..9), A343615 (P_{3,2}(5): same for p==2 (mod 3)).

Cf. A085965 (PrimeZeta(5)), A002476 (primes of the form 3k+1).

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 *)

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.

A343607_upto(N=100)={localprec(N+5);digits((PrimeZeta31(5)+1)\.1^N)[^1]} \\ cf. A175644 for PrimeZeta31

A343615 Decimal expansion of P_{3,2}(5) = Sum 1/p^5 over primes == 2 (mod 3).

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Apr 22 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.

			0.0315771357190039419560337803437163963477729963832486145790258341228297557...

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=5) on p. 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A003627 (primes 3k-1), A000584 (n^5), A085965 (PrimeZeta(5)), A021247 (1/3^5).

Cf. A343625 (same for primes 3k+1), A086035 (for primes 4k+1), A085994 (for primes 4k+3), A343612 - A343619 (P_{3,2}(2..9): same for 1/p^2, ..., 1/p^9).

s=0;forprimestep(p=2,1e8,3,s+=1./p^4);s \\ For illustration: using primes up to 10^N gives about 3N+2 (= 26 for N=8) correct digits.

A343615_upto(N=100, s=5)={localprec(N+5); digits((sumeulerrat(1/p^s)-1/3^s-PrimeZeta31(s)+1)\.1^N)[^1]} \\ see A175644 for the function PrimeZeta31, A343612 for a function PrimeZeta32

P_{3,2}(5) = P(5) - 1/3^5 - P_{3,1}(5).

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A085965 Decimal expansion of the prime zeta function at 5.

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A085994 Decimal expansion of the prime zeta modulo function at 5 for primes of the form 4k+3.

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A343625 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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A343615 Decimal expansion of P_{3,2}(5) = Sum 1/p^5 over primes == 2 (mod 3).

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