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 A328895 #41 Feb 16 2025 08:33:58
%S A328895 8,7,2,3,5,8,0,2,4,9,5,4,8,5,9,9,4,1,7,6,9,6,9,5,1,1,7,0,2,1,1,7,5,6,
%T A328895 6,1,2,3,9,9,8,3,2,8,3,8,6,8,5,0,5,2,9,5,7,6,9,1,8,7,0,8,3,4,3,9,9,8,
%U A328895 8,4,7,0,3,5,4,1,3,4,6,5,1,8,3,3,4,2,5,1,6,7,1
%N A328895 Decimal expansion of Sum_{k>=1} Kronecker(8,k)/k^2.
%C A328895 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 A328895 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 A328895 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 A328895 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 A328895 In this sequence we have Chi = A091337 and s = 2.
%H A328895 M. W. Coffey, <a href="https://arxiv.org/abs/1701.07064">Summatory relations and prime products for the Stieltjes constants and other related results</a>, arXiv:1701.07064 [math.NT], 2017, eq. (2.1).
%H A328895 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=2.
%H A328895 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DirichletL-Series.html">Dirichlet L-Series</a>.
%H A328895 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PolygammaFunction.html">Polygamma Function</a>.
%F A328895 Equals Pi^2/(8*sqrt(2)).
%F A328895 Equals (zeta(2,1/8) - zeta(2,3/8) - zeta(2,5/8) + zeta(2,7/8))/64, where zeta(s,a) is the Hurwitz zeta function.
%F A328895 Equals (polylog(2,u) - polylog(2,u^3) - polylog(2,-u) + polylog(2,-u^3))/sqrt(8), where u = sqrt(2)/2 + i*sqrt(2)/2 is an 8th primitive root of unity, i = sqrt(-1).
%F A328895 Equals (polygamma(1,1/8) - polygamma(1,3/8) - polygamma(1,5/8) + polygamma(1,7/8))/64.
%F A328895 Equals -Integral_{x=0..oo} log(x)/(x^4 + 1) dx. - _Amiram Eldar_, Jul 17 2020
%F A328895 Equals 1/(Product_{p prime == 1 or 7 (mod 8)} (1 - 1/p^2) * Product_{p prime == 3 or 5 (mod 8)} (1 + 1/p^2)). - _Amiram Eldar_, Dec 17 2023
%e A328895 1 - 1/3^2 - 1/5^2 + 1/7^2 + 1/9^2 - 1/11^2 - 1/13^2 + 1/15^2 + ... = Pi^2/(8*sqrt(2)) = 0.8723580249...
%t A328895 RealDigits[Pi^2/(8*Sqrt[2]), 10, 102] // First
%o A328895 (PARI) default(realprecision, 100); Pi^2/(8*sqrt(2))
%Y A328895 Cf. A091337, A346728, A346727, A346260.
%Y A328895 Decimal expansion of Sum_{k>=1} Kronecker(d,k)/k^2, where d is a fundamental discriminant: A309710 (d=-8), A103133 (d=-7), A006752 (d=-4), A086724 (d=-3), A013661 (d=1), A328717 (d=5), this sequence (d=8), A258414 (d=12).
%Y A328895 Decimal expansion of Sum_{k>=1} Kronecker(8,k)/k^s: A196525 (s=1), this sequence (s=2), A329715 (s=3).
%K A328895 nonn,cons
%O A328895 0,1
%A A328895 _Jianing Song_, Nov 19 2019