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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).

%I A086239 #55 Feb 16 2025 08:32:50
%S A086239 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,
%T A086239 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,
%U A086239 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
%N 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).
%C A086239 This is Sum_{p prime, p>=3} -(-4/p)/p where (-4/.) is the Legendre symbol and is equal to - L(1,(-4/.)) plus an absolutely convergent sum (and therefore converges).
%D A086239 Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
%D A086239 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, pp. 94-98.
%H A086239 Julien Benney, Mark Underwood, Andrew J. Walker and David Broadhurst, <a href="/A086239/a086239.txt">Is this a convergent series and if so what is its sum?</a>, digest of 12 messages in primenumbers Yahoo group, Oct 26 - Oct 30, 2009. [Cached copy]
%H A086239 David Broadhurst, <a href="http://groups.yahoo.com/group/primenumbers/message/21083">post in primenumbers group</a>, Oct 29 2009. [Broken link]
%H A086239 Henri Cohen, <a href="http://www.math.u-bordeaux.fr/~cohen/hardylw.dvi">High Precision Computation of Hardy-Littlewood Constants</a>, (1991).
%H A086239 Henri Cohen, <a href="/A221712/a221712.pdf">High-precision computation of Hardy-Littlewood constants</a>. [pdf copy, with permission]
%H A086239 David Dummit, Andrew Granville, and Hershy Kisilevsky, <a href="https://doi.org/10.1112/S0025579315000339">Big biases amongst products of two primes</a>, Mathematika 62 (2016), pp. 502-507; <a href="https://arxiv.org/abs/1411.4594">arXiv preprint</a>, arXiv:1411.4594 [math.NT], 2014.
%H A086239 Richard J. Mathar, <a href="https://arxiv.org/abs/1008.2547">Table of Dirichlet L-series and prime zeta modulo functions for small moduli</a>, arXiv:1008.2547 [math.NT], 2010-2015, variable S(m=4,r=2,s=1) Section 3.1.
%H A086239 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeSums.html">Prime Sums</a>.
%F A086239 Equals A368646 - A368645. - _Amiram Eldar_, Jan 02 2024
%e A086239 0.33498132529999...
%t A086239 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 *)
%t A086239 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 *)
%o A086239 (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
%Y A086239 Cf. A166509, A368645, A368646.
%K A086239 nonn,cons
%O A086239 0,1
%A A086239 _Eric W. Weisstein_, Jul 13 2003
%E A086239 Edited by _N. J. A. Sloane_, Jun 10 2008
%E A086239 Corrected a(9) and example, added a(10)-a(104) following Broadhurst and Cohen. - _M. F. Hasler_, Oct 29 2009