A086239 Decimal expansion of Sum_{k>=2} c(k)/prime(k), where c(k) = -1 if p == 1 (mod 4) and c(k) = +1 if p == 3 (mod 4).
3, 3, 4, 9, 8, 1, 3, 2, 5, 2, 9, 9, 9, 9, 3, 1, 8, 1, 0, 6, 3, 3, 1, 7, 1, 2, 1, 4, 8, 7, 5, 4, 3, 5, 7, 3, 7, 7, 9, 9, 7, 5, 3, 8, 0, 7, 5, 5, 0, 7, 7, 0, 4, 8, 1, 0, 8, 0, 2, 0, 5, 7, 8, 8, 4, 5, 2, 2, 2, 8, 4, 3, 2, 7, 1, 8, 8, 4, 1, 1, 0, 6, 2, 4, 8, 9, 9, 6, 3, 1, 0, 2, 9, 8, 0, 3, 3, 4, 5, 3, 9, 2, 4, 8, 6
Offset: 0
Examples
0.33498132529999...
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.
Links
- Julien Benney, Mark Underwood, Andrew J. Walker and David Broadhurst, Is this a convergent series and if so what is its sum?, digest of 12 messages in primenumbers Yahoo group, Oct 26 - Oct 30, 2009. [Cached copy]
- David Broadhurst, post in primenumbers group, Oct 29 2009. [Broken link]
- Henri Cohen, High Precision Computation of Hardy-Littlewood Constants, (1991).
- Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
- David Dummit, Andrew Granville, and Hershy Kisilevsky, Big biases amongst products of two primes, Mathematika 62 (2016), pp. 502-507; arXiv preprint, arXiv:1411.4594 [math.NT], 2014.
- Richard J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015, variable S(m=4,r=2,s=1) Section 3.1.
- Eric Weisstein's World of Mathematics, Prime Sums.
Programs
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Mathematica
Do[Print[N[Log[2]/2 + Sum[Log[2^(4*n)*(2^(2*n + 1) + 1)*(2^(2*n + 3) - 4)*(Zeta[4*n + 2] / (Zeta[2*n + 1, 1/4] - Zeta[2*n + 1, 3/4])^2)] * MoebiusMu[2*n + 1]/(4*n + 2), {n, 1, m}], 120]], {m, 20, 200, 20}] (* Vaclav Kotesovec, Jun 28 2020 *) S[m_, n_, s_] := (t = 1; sums = 0; difs = 1; While[Abs[difs] > 10^(-digits - 5) || difs == 0, difs = (MoebiusMu[t]/t) * Log[If[s*t == 1, DirichletL[m, n, s*t], Sum[Zeta[s*t, j/m]*DirichletCharacter[m, n, j]^t, {j, 1, m}]/m^(s*t)]]; sums = sums + difs; t++]; sums); $MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[-S[4, 2, 1], digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 22 2021 *) -
PARI
/* the given number of primes and terms in the sum yield over 105 correct digits */ P=vector(15, k, (2-prime(k)%4)/prime(k)); -sum(s=1,60, moebius(s)/s*log( prod( k=2, #P, 1-P[k]^s, if(s%2, if(s==1, Pi/4, sumalt(k=0,(-1)^k/(2*k+1)^s)),zeta(s)*(1-1/2^s) ))), sum(k=2,#P, P[k], .)) \\ M. F. Hasler, Oct 29 2009
Formula
Extensions
Edited by N. J. A. Sloane, Jun 10 2008
Corrected a(9) and example, added a(10)-a(104) following Broadhurst and Cohen. - M. F. Hasler, Oct 29 2009
Comments