This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A329715 #17 Feb 16 2025 08:33:58
%S A329715 9,5,8,3,8,0,4,5,4,5,6,3,0,9,4,5,6,2,0,5,1,6,6,9,4,0,2,8,6,1,5,7,7,8,
%T A329715 1,8,8,2,4,8,9,5,3,1,7,9,3,9,7,7,5,3,4,0,7,5,7,5,0,4,5,0,7,0,4,7,0,7,
%U A329715 5,6,9,7,4,8,4,2,9,7,9,3,6,4,7,8,2,5,2,6,9,9,7
%N A329715 Decimal expansion of Sum_{k>=1} Kronecker(8,k)/k^3.
%C A329715 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 A329715 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 A329715 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 A329715 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 A329715 In this sequence we have Chi = A091337 and s = 3.
%H A329715 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. 99.
%H A329715 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>, Section 2.2 at m=8, r=2, s=3.
%H A329715 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DirichletL-Series.html">Dirichlet L-Series</a>.
%H A329715 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PolygammaFunction.html">Polygamma Function</a>.
%F A329715 Equals (zeta(3,1/8) - zeta(3,3/8) - zeta(3,5/8) + zeta(3,7/8))/512, where zeta(s,a) is the Hurwitz zeta function.
%F A329715 Equals (polylog(3,u) - polylog(3,u^3) - polylog(3,-u) + polylog(3,-u^3))/sqrt(8), where u = sqrt(2)/2 + i*sqrt(2)/2 is an 8th primitive root of unity, i = sqrt(-1).
%F A329715 Equals (polygamma(2,1/8) - polygamma(2,3/8) - polygamma(2,5/8) + polygamma(2,7/8))/(-1024).
%F A329715 Equals 1/(Product_{p prime == 1 or 7 (mod 8)} (1 - 1/p^3) * Product_{p prime == 3 or 5 (mod 8)} (1 + 1/p^3)). - _Amiram Eldar_, Dec 17 2023
%e A329715 1 - 1/3^3 - 1/5^3 + 1/7^3 + 1/9^3 - 1/11^3 - 1/13^3 + 1/15^3 + ... = 0.9583804545...
%t A329715 (PolyGamma[2, 1/8] - PolyGamma[2, 3/8] - PolyGamma[2, 5/8] + PolyGamma[2, 7/8])/(-1024) // RealDigits[#, 10, 102] & // First
%Y A329715 Cf. A091337.
%Y A329715 Decimal expansion of Sum_{k>=1} Kronecker(d,k)/k^3, where d is a fundamental discriminant: A251809 (d=-8), A327135 (d=-7), A153071 (d=-4), A129404 (d=-3), A002117 (d=1), A328723 (d=5), this sequence (d=8), A329716 (d=12).
%Y A329715 Decimal expansion of Sum_{k>=1} Kronecker(8,k)/k^s: A196525 (s=1), A328895 (s=2), this sequence (s=3).
%K A329715 nonn,cons
%O A329715 0,1
%A A329715 _Jianing Song_, Nov 19 2019