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A085969 Decimal expansion of the prime zeta function at 9. 25

%I #52 Feb 23 2024 07:28:00

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

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

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

%N Decimal expansion of the prime zeta function at 9.

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

%D Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.

%D J. W. L. Glaisher, On the Sums of Inverse Powers of the Prime Numbers, Quart. J. Math. 25, 347-362, 1891.

%H Jason Kimberley, <a href="/A085969/b085969.txt">Table of n, a(n) for n = -2..1999</a>

%H Henri Cohen, <a href="http://www.math.u-bordeaux.fr/~cohen/hardylw.dvi">High Precision Computation of Hardy-Littlewood Constants</a>, Preprint, 1998.

%H Henri Cohen, <a href="/A221712/a221712.pdf">High-precision computation of Hardy-Littlewood constants</a>. [pdf copy, with permission]

%H X. Gourdon and P. Sebah, <a href="http://numbers.computation.free.fr/Constants/Miscellaneous/constantsNumTheory.html">Some Constants from Number theory</a>

%H R. J. Mathar, <a href="http://arxiv.org/abs/0803.0900">Series of reciprocal powers of k-almost primes</a>, arXiv:0803.0900 [math.NT], 2008-2009. Table 1.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeZetaFunction.html">Prime Zeta Function</a>

%F P(9) = Sum_{p prime} 1/p^9 = Sum_{n=1..inf} mobius(n)*log(zeta(9*n))/n.

%F Equals Sum_{k>=1} 1/A179665(k). - _Amiram Eldar_, Jul 27 2020

%e 0.0020044675749624506630...

%t pz9[n_] := pz9[n] = Join[{0, 0}, Sum[ MoebiusMu[k]*Log[Zeta[9*k]]/k, {k, 1, n}] // RealDigits[#, 10, 103]& // First]; pz9[100]; pz9[n = 200]; While[pz9[n] != pz9[n - 100], n = n + 100]; pz9[n] (* _Jean-François Alcover_, Feb 14 2013, from formula *)

%t RealDigits[ PrimeZetaP[ 9], 10, 111][[1]] (* _Robert G. Wilson v_, Sep 03 2014 *)

%o (Magma) R := RealField(106);

%o PrimeZeta := func<k,N | &+[R|MoebiusMu(n)/n*Log(ZetaFunction(R,k*n)): n in[1..N]]>;

%o [0,0] cat Reverse(IntegerToSequence(Floor(PrimeZeta(9,38)*10^105)));

%o // _Jason Kimberley_, Dec 30 2016

%o (PARI) sumeulerrat(1/p, 9) \\ _Hugo Pfoertner_, Feb 03 2020

%Y Decimal expansion of the prime zeta function: A085548 (at 2), A085541 (at 3), A085964 (at 4) to A085968 (at 8), this sequence (at 9).

%Y Cf. A013667, A179665.

%K cons,easy,nonn

%O -2,1

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

%E Changed offset and adapted data by _Hugo Pfoertner_, Jan 31 2020

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Last modified March 28 22:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)