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%I #99 Mar 12 2024 02:44:51
%S 4,5,2,2,4,7,4,2,0,0,4,1,0,6,5,4,9,8,5,0,6,5,4,3,3,6,4,8,3,2,2,4,7,9,
%T 3,4,1,7,3,2,3,1,3,4,3,2,3,9,8,9,2,4,2,1,7,3,6,4,1,8,9,3,0,3,5,1,1,6,
%U 5,0,2,7,3,6,3,9,1,0,8,7,4,4,4,8,9,5,7,5,4,4,3,5,4,9,0,6,8,5,8,2,2,2,8,0,6
%N Decimal expansion of the prime zeta function at 2: Sum_{p prime} 1/p^2.
%C Mathar's Table 1 (cited below) lists expansions of the prime zeta function at integers s in 10..39. - _Jason Kimberley_, Jan 05 2017
%D Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
%D Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, pp. 94-98.
%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="/A085548/b085548.txt">Table of n, a(n) for n = 0..1093</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 Persi Diaconis, Frederick Mosteller, and Hironari Onishi, <a href="http://dx.doi.org/10.1016/0022-314X(77)90022-1">Second-order terms for the variances and covariances of the number of prime factors-including the square free case</a>, J. Number Theory 9 (1977), no. 2, 187--202. MR0434991 (55 #7953).
%H Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 171 and 190.
%H Xavier Gourdon and Pascal Sebah, <a href="http://numbers.computation.free.fr/Constants/Miscellaneous/constantsNumTheory.html">Some Constants from Number theory</a>.
%H Shanta Laishram and Florian Luca, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Laishram/lai2.html">Rectangles Of Nonvisible Lattice Points</a>, J. Int. Seq. 18 (2015), Article 15.10.8, Theorem 1.
%H Jon Lee, Joseph Paat, Ingo Stallknecht, and Luze Xu, <a href="https://arxiv.org/abs/2105.08160">Polynomial upper bounds on the number of differing columns of Delta-modular integer programs</a>, arXiv:2105.08160 [math.OC], 2021, see page 23.
%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 Gerhard Niklasch and Pieter Moree, <a href="/A001692/a001692.html">Some number-theoretical constants</a>. [Cached copy]
%H Michael Ian Shamos, <a href="http://euro.ecom.cmu.edu/people/faculty/mshamos/PST.pdf">Property Enumerators and a Partial Sum Theorem</a>, 2011; <a href="https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=e503bec3c04c3f94cb267882724dd414e143141b">alternative link</a>.
%H Hanson Smith, <a href="https://arxiv.org/abs/1808.04809">Ramification in the Division Fields of Elliptic Curves and an Application to Sporadic Points on Modular Curves</a>, arXiv:1810.04809 [math.NT], 2018.
%H Hanson Smith, <a href="http://math.colorado.edu/~hwsmith/Ram%20Div%20Formal%20Group%2038.pdf">Ramification in Division Fields and Sporadic Points on Modular Curves</a>, U. Conn. (2020).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DistinctPrimeFactors.html">Distinct Prime Factors</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeSums.html">Prime Sums</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeZetaFunction.html">Prime Zeta Function</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Prime_zeta_function">Prime Zeta Function</a>.
%F P(2) = Sum_{p prime} 1/p^2 = Sum_{n>=1} mobius(n)*log(zeta(2*n))/n. - Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003
%F Equals A085991 + A086032 + 1/4. - _R. J. Mathar_, Jul 22 2010
%F Equals Sum_{k>=1} 1/A001248(k). - _Amiram Eldar_, Jul 27 2020
%F Equals Sum_{k>=2} pi(k)*(2*k+1)/(k^2*(k+1)^2), where pi(k) = A000720(k) (Shamos, 2011, p. 9). - _Amiram Eldar_, Mar 12 2024
%e 0.4522474200410654985065... = 1/2^2 + 1/3^2 + 1/5^2 +1/7^2 + 1/11^2 + 1/13^2 + ...
%t RealDigits[PrimeZetaP[2], 10, 105][[1]] (* _Jean-François Alcover_, Jun 24 2011, updated May 06 2021 *)
%o (PARI) recip2(n) = { v=0; p=1; forprime(y=2,n, v=v+1./y^2; ); print(v) }
%o (PARI) eps()=my(p=default(realprecision)); precision(2.>>(32*ceil(p*38539962/371253907)),9)
%o lm=lambertw(log(4)/eps())\log(4);
%o sum(k=1,lm, moebius(k)/k*log(abs(zeta(2*k)))) \\ _Charles R Greathouse IV_, Jul 19 2013
%o (PARI) sumeulerrat(1/p,2) \\ _Hugo Pfoertner_, Feb 03 2020
%o (Magma) R := RealField(106);
%o PrimeZeta := func<k, N |&+[R|MoebiusMu(n)/n*Log(ZetaFunction(R,k*n)):n in[1..N]]>;
%o Reverse(IntegerToSequence(Floor(PrimeZeta(2,173)*10^105)));
%o // _Jason Kimberley_, Dec 30 2016
%Y Decimal expansion of the prime zeta function: this sequence (at 2), A085541 (at 3), A085964 (at 4) to A085969 (at 9).
%Y Cf. A136271 (derivative), A117543 (semiprimes), A222056, A209329, A124012.
%Y Cf. A000720, A001248, A013661, A078437, A242301.
%K easy,nonn,cons
%O 0,1
%A _Cino Hilliard_, Jul 03 2003
%E More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003
%E Offset corrected by _R. J. Mathar_, Feb 05 2009
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