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A085968 Decimal expansion of the prime zeta function at 8. 5
0, 0, 4, 0, 6, 1, 4, 0, 5, 3, 6, 6, 5, 1, 7, 8, 3, 0, 5, 6, 0, 5, 2, 3, 4, 3, 9, 1, 4, 2, 6, 8, 3, 0, 8, 0, 5, 2, 2, 9, 7, 7, 1, 4, 4, 5, 1, 2, 0, 7, 1, 7, 4, 1, 0, 0, 1, 0, 3, 2, 6, 8, 8, 6, 8, 1, 7, 2, 8, 6, 3, 0, 4, 0, 7, 0, 7, 8, 8, 0, 4, 4, 0, 6, 0, 9, 2, 2, 8, 2, 8, 0, 5, 3, 0, 4, 3, 1, 3, 4, 4, 2, 6, 5, 6 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

REFERENCES

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..1971

H. Cohen, High Precision Computation of Hardy-Littlewood Constants, Preprint.

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(8) = Sum_{p prime>=2} 1/p^8 = Sum_{n>=1} mobius(n)*log(zeta(8*n))/n.

EXAMPLE

0.0040614053665178305605...

MAPLE

A085968:= proc(i) print(evalf(add(1/ithprime(k)^8, k=1..i), 100)); end:

A085968(100000); # Paolo P. Lava, May 29 2012

MATHEMATICA

s[n_] := s[n] = Sum[ MoebiusMu[k]*Log[Zeta[8*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[ 8], 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, 0] cat Reverse(IntegerToSequence(Floor(PrimeZeta(8, 43)*10^105)));

// Jason Kimberley, Dec 30 2016

CROSSREFS

Decimal expansion of the prime zeta function: A085548 (at 2), A085541 (at 3), A085964 (at 4) to A085967 (at 7), this sequence (at 8), A085969 (at 9).

Cf. A013666.

Sequence in context: A132953 A195207 A157721 * A010637 A200692 A127447

Adjacent sequences:  A085965 A085966 A085967 * A085969 A085970 A085971

KEYWORD

cons,easy,nonn

AUTHOR

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

STATUS

approved

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Last modified January 23 01:55 EST 2018. Contains 298093 sequences.