A309710 - OEIS

%I #40 Dec 17 2024 10:24:42

%S 1,0,6,4,7,3,4,1,7,1,0,4,3,5,0,3,3,7,0,3,9,2,8,2,7,4,5,1,4,6,1,6,6,8,

%T 8,8,9,4,8,3,0,9,9,1,5,1,7,7,4,4,8,5,1,2,4,4,1,9,8,7,4,5,0,8,0,6,3,9,

%U 9,0,1,7,1,7,5,8,6,4,3,7,6,3,6,6,6,5,3,4,2,5,0

%N Decimal expansion of Sum_{k>=1} Kronecker(-8,k)/k^2.

%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 = A188510 and s = 2.

%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. 98.

%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>, Section 2.2 L(m=8, r=4, s=2).

%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 (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 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 Equals (polygamma(1,1/8) + polygamma(1,3/8) - polygamma(1,5/8) - polygamma(1,7/8))/64.

%F Equals 1/(Product_{p prime == 1 or 3 (mod 8)} (1 - 1/p^2) * Product_{p prime == 5 or 7 (mod 8)} (1 + 1/p^2)). - _Amiram Eldar_, Dec 17 2023

%e 1 + 1/3^2 - 1/5^2 - 1/7^2 + 1/9^2 + 1/11^2 - 1/13^2 - 1/15^2 + ...= 1.0647341710...

%t (PolyGamma[1, 1/8] + PolyGamma[1, 3/8] - PolyGamma[1, 5/8] - PolyGamma[1, 7/8])/64 // RealDigits[#, 10, 102] & // First

%Y Cf. A188510.

%Y Decimal expansion of Sum_{k>=1} Kronecker(d,k)/k^2, where d is a fundamental discriminant: this sequence (d=-8), A103133 (d=-7), A006752 (d=-4), A086724 (d=-3), A013661 (d=1), A328717 (d=5), A328895 (d=8), A258414 (d=12).

%Y Decimal expansion of Sum_{k>=1} Kronecker(-8,k)/k^s: A093954 (s=1), this sequence (s=2), A251809 (s=3).

%K nonn,cons

%O 1,3

%A _Jianing Song_, Nov 19 2019