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A085964 Decimal expansion of the prime zeta function at 4. 19
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

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

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(4) = Sum_{p prime>=2} 1/p^4 = Sum_{n>=1} mobius(n)*log(zeta(4*n))/n

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

EXAMPLE

0.0769931397642468449426...

MAPLE

A085964:= proc(i) print(evalf(add(1/ithprime(k)^4, k=1..i), 100)); end:

A085964(100000); # Paolo P. Lava, May 29 2012

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-Fran├žois 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

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.

Sequence in context: A201766 A197588 A021569 * A281313 A082121 A215338

Adjacent sequences:  A085961 A085962 A085963 * A085965 A085966 A085967

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 23:54 EST 2017. Contains 281222 sequences.