A085548 Decimal expansion of the prime zeta function at 2: Sum_{p prime} 1/p^2.

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, 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, 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

Offset

0,1

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.

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, pp. 94-98.

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..1093

Henri Cohen, High Precision Computation of Hardy-Littlewood Constants, Preprint, 1998.

Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]

Persi Diaconis, Frederick Mosteller, and Hironari Onishi, Second-order terms for the variances and covariances of the number of prime factors-including the square free case, J. Number Theory 9 (1977), no. 2, 187--202. MR0434991 (55 #7953).

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 171 and 190.

Xavier Gourdon  and  Pascal Sebah, Some Constants from Number theory.

Shanta Laishram and Florian Luca, Rectangles Of Nonvisible Lattice Points, J. Int. Seq. 18 (2015), Article 15.10.8, Theorem 1.

Jon Lee, Joseph Paat, Ingo Stallknecht, and Luze Xu, Polynomial upper bounds on the number of differing columns of Delta-modular integer programs, arXiv:2105.08160 [math.OC], 2021, see page 23.

R. J. Mathar, Series of reciprocal powers of k-almost primes, arXiv:0803.0900 [math.NT], 2008-2009. Table 1.

Gerhard Niklasch and Pieter Moree, Some number-theoretical constants. [Cached copy]

Michael Ian Shamos, Property Enumerators and a Partial Sum Theorem, 2011; alternative link.

Hanson Smith, Ramification in the Division Fields of Elliptic Curves and an Application to Sporadic Points on Modular Curves, arXiv:1810.04809 [math.NT], 2018.

Hanson Smith, Ramification in Division Fields and Sporadic Points on Modular Curves, U. Conn. (2020).

Eric Weisstein's World of Mathematics, Distinct Prime Factors.

Eric Weisstein's World of Mathematics, Prime Sums.

Eric Weisstein's World of Mathematics, Prime Zeta Function.

Wikipedia, Prime Zeta Function.

Formula

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

Equals A085991 + A086032 + 1/4. - R. J. Mathar, Jul 22 2010

Equals Sum_{k>=1} 1/A001248(k). - Amiram Eldar, Jul 27 2020

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

Example

0.4522474200410654985065... = 1/2^2 + 1/3^2 + 1/5^2 +1/7^2 + 1/11^2 + 1/13^2 + ...

Mathematica

RealDigits[PrimeZetaP[2], 10, 105][[1]]  (* Jean-François Alcover, Jun 24 2011, updated May 06 2021 *)

Prog

(PARI) recip2(n) = { v=0; p=1; forprime(y=2, n, v=v+1./y^2; ); print(v) }
(PARI) eps()=my(p=default(realprecision)); precision(2.>>(32*ceil(p*38539962/371253907)), 9)
lm=lambertw(log(4)/eps())\log(4);
sum(k=1, lm, moebius(k)/k*log(abs(zeta(2*k)))) \\ Charles R Greathouse IV, Jul 19 2013
(PARI) sumeulerrat(1/p, 2) \\ Hugo Pfoertner, Feb 03 2020
(Magma) R := RealField(106);
PrimeZeta := func<k, N |&+[R|MoebiusMu(n)/n*Log(ZetaFunction(R, k*n)):n in[1..N]]>;
Reverse(IntegerToSequence(Floor(PrimeZeta(2, 173)*10^105)));
// Jason Kimberley, Dec 30 2016

Crossrefs

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

Cf. A136271 (derivative), A117543 (semiprimes), A222056, A209329, A124012.

Cf. A000720, A001248, A013661, A078437, A242301.

Sequence in context: A016715 A337192 A255701 * A329957 A074459 A371747

Adjacent sequences: A085545 A085546 A085547 * A085549 A085550 A085551

Keyword

easy,nonn,cons

Author

Cino Hilliard, Jul 03 2003

Extensions

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

Offset corrected by R. J. Mathar, Feb 05 2009

Status

approved