A085964 - OEIS

A085964

Decimal expansion of the prime zeta function at 4.

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Mathar's Table 1 (cited below) lists expansions of the prime zeta function at integers s in 10..39. - Jason Kimberley, Jan 05 2017

REFERENCES

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.

LINKS

Jason Kimberley, Table of n, a(n) for n = 0..1603

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

FORMULA

P(4) = Sum_{p prime} 1/p^4 = Sum_{n>=1} mobius(n)*log(zeta(4*n))/n

Equals A086034 + A085993 + 1/16. - R. J. Mathar, Jul 22 2010

Equals Sum_{k>=1} 1/A030514(k). - Amiram Eldar, Jul 27 2020

EXAMPLE

0.0769931397642468449426...

MATHEMATICA

s[n_] := s[n] = Sum[ MoebiusMu[k]*Log[Zeta[4*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[ 4], 10, 111][[1]] (* Robert G. Wilson v, Sep 03 2014 *)

PROG

(Magma) R := RealField(106);

PrimeZeta := func<k, N|

&+[R|MoebiusMu(n)/n*Log(ZetaFunction(R, k*n)):n in[1..N]]>;

[0]cat Reverse(IntegerToSequence(Floor(PrimeZeta(4, 87)*10^105)));

// Jason Kimberley, Dec 30 2016

(PARI) sumeulerrat(1/p, 4) \\ Hugo Pfoertner, Feb 03 2020

CROSSREFS

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

Cf. A013662, A030514, A242303.

Sequence in context: A307951 A197588 A021569 * A281313 A082121 A309622