|
%I #46 Dec 17 2023 03:06:56
%S 1,0,9,3,3,4,3,0,6,9,4,2,9,5,3,3,5,7,1,9,7,6,5,7,9,8,1,5,0,0,7,7,0,0,
%T 2,3,4,7,8,0,1,9,2,5,8,4,8,3,2,3,8,3,6,4,6,3,5,0,2,3,0,9,4,3,2,4,3,2,
%U 8,1,0,6,9,0,3,2,3,6,2,1,7,4,3,4,0,4,6,2,2,9,2
%N Decimal expansion of Sum_{k>=1} Kronecker(-7,k)/k^3.
%C Let Chi() be a primitive character modulo d, the so-called Dirichlet L-series L(s,Chi) is the analytic continuation (see the functional equations involving L(s,Chi) in the MathWorld link entitled Dirichlet L-Series) of the sum Sum_{k>=1} Chi(k)/k^s, Re(s)>0 (if d = 1, the sum converges requires Re(s)>1).
%C If s != 1, we can represent L(s,Chi) in terms of the Hurwitz zeta function by L(s,Chi) = (Sum_{k=1..d} Chi(k)*zeta(s,k/d))/d^s.
%C L(s,Chi) can also be represented in terms of the polylog function by L(s,Chi) = (Sum_{k=1..d} Chi'(k)*polylog(s,u^k))/(Sum_{k=1..d} Chi'(k)*u^k), where Chi' is the complex conjugate of Chi, u is any primitive d-th root of unity.
%C If m is a positive integer, we have L(m,Chi) = (Sum_{k=1..d} Chi(k)*polygamma(m-1,k/d))/((-d)^m*(m-1)!).
%C In this sequence we have Chi = A175629 and s = 3.
%H R. 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, L(m=7,r=4,s=3).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DirichletL-Series.html">Dirichlet L-Series</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PolygammaFunction.html">Polygamma Function</a>.
%F Equals 32*Pi^3/(343*sqrt(7)).
%F Equals (zeta(3,1/7) + zeta(3,2/7) - zeta(3,3/7) + zeta(3,4/7) - zeta(3,5/7) - zeta(3,6/7))/343.
%F Equals (polylog(3,u) + polylog(3,u^2) - polylog(3,u^3) + polylog(3,u^4) - polylog(3,u^5) - polylog(3,u^6))/sqrt(-7), where u = exp(2*Pi*i/7) is a 7th primitive root of unity, i = sqrt(-1).
%F Equals (polygamma(2,1/7) + polygamma(2,2/7) - polygamma(2,3/7) + polygamma(2,4/7) - polygamma(2,5/7) - polygamma(2,6/7))/(-686).
%F Equals 1/(Product_{p prime == 1, 2 or 4 (mod 7)} (1 - 1/p^3) * Product_{p prime == 3, 5 or 6 (mod 7)} (1 + 1/p^3)). - _Amiram Eldar_, Dec 17 2023
%e 1 + 1/2^3 - 1/3^3 + 1/4^3 - 1/5^3 - 1/6^3 + 1/8^3 + 1/9^3 - 1/10^3 + 1/11^3 - 1/12^3 - 1/13^3 + ... = 32*Pi^3/(343*sqrt(7)) = 1.0933430694...
%t RealDigits[32*Pi^3/(343*Sqrt[7]), 10, 102] // First
%o (PARI) default(realprecision, 100); 32*Pi^3/(343*sqrt(7))
%Y Cf. A175629.
%Y Decimal expansion of Sum_{k>=1} Kronecker(d,k)/k^3, where d is a fundamental discriminant: A251809 (d=-8), this sequence (d=-7), A153071 (d=-4), A129404 (d=-3), A002117 (d=1), A328723 (d=5), A329715 (d=8), A329716 (d=12).
%Y Decimal expansion of Sum_{k>=1} Kronecker(-7,k)/k^s: A326919 (s=1), A103133 (s=2), this sequence (s=3).
%K nonn,cons
%O 1,3
%A _Jianing Song_, Nov 19 2019
|
