A329716 - OEIS

%I #16 Feb 16 2025 08:33:58

%S 9,9,0,0,4,0,0,1,9,4,3,8,1,5,9,9,4,9,7,9,1,8,1,6,7,7,6,8,6,3,3,0,4,0,

%T 5,0,8,5,6,8,8,5,0,6,7,6,5,7,2,3,6,1,4,5,5,5,3,6,6,0,7,0,0,3,4,2,3,5,

%U 2,0,5,3,3,6,7,1,8,1,1,6,7,7,8,5,6,0,2,2,3,1,8

%N Decimal expansion of Sum_{k>=1} Kronecker(12,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 = A110161 and s = 3.

%H 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 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DirichletL-Series.html">Dirichlet L-Series</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PolygammaFunction.html">Polygamma Function</a>.

%F Equals (zeta(3,1/12) - zeta(3,5/12) - zeta(3,7/12) + zeta(3,11/12))/1728, where zeta(s,a) is the Hurwitz zeta function.

%F Equals (polylog(3,u) - polylog(3,u^5) - polylog(3,-u) + polylog(3,-u^5))/sqrt(12), where u = (sqrt(3)+i)/2 is a 12th primitive root of unity, i = sqrt(-1).

%F Equals (polygamma(2,1/12) - polygamma(2,5/12) - polygamma(2,7/12) + polygamma(2,11/12))/(-3456).

%F Equals 1/(Product_{p prime == 1 or 11 (mod 12)} (1 - 1/p^3) * Product_{p prime == 5 or 7 (mod 12)} (1 + 1/p^3)). - _Amiram Eldar_, Dec 17 2023

%e 1 - 1/5^3 - 1/7^3 + 1/11^3 + 1/13^3 - 1/17^3 - 1/19^3 + 1/23^3 + ... = 0.9900400194...

%t (PolyGamma[2, 1/12] - PolyGamma[2, 5/12] - PolyGamma[2, 7/12] + PolyGamma[2, 11/12])/(-3456) // RealDigits[#, 10, 102] & // First

%Y Cf. A110161.

%Y 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), A329715 (d=8), this sequence (d=12).

%Y Decimal expansion of Sum_{k>=1} Kronecker(12,k)/k^s: A196530 (s=1), A258414 (s=2), this sequence (s=3).

%K nonn,cons

%O 0,1

%A _Jianing Song_, Nov 19 2019