A328895 Decimal expansion of Sum_{k>=1} Kronecker(8,k)/k^2.

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

Offset

0,1

Comments

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).

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.

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.

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)!).

In this sequence we have Chi = A091337 and s = 2.

Links

Table of n, a(n) for n=0..90.

M. W. Coffey, Summatory relations and prime products for the Stieltjes constants and other related results, arXiv:1701.07064 [math.NT], 2017, eq. (2.1).

R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, Section 2.2 at m=8, r=2, s=2.

Eric Weisstein's World of Mathematics, Dirichlet L-Series.

Eric Weisstein's World of Mathematics, Polygamma Function.

Formula

Equals Pi^2/(8*sqrt(2)).

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.

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).

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

Equals -Integral_{x=0..oo} log(x)/(x^4 + 1) dx. - Amiram Eldar, Jul 17 2020

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

Example

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...

Mathematica

RealDigits[Pi^2/(8*Sqrt[2]), 10, 102] // First

Prog

(PARI) default(realprecision, 100); Pi^2/(8*sqrt(2))

Crossrefs

Cf. A091337, A346728, A346727, A346260.

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).

Decimal expansion of Sum_{k>=1} Kronecker(8,k)/k^s: A196525 (s=1), this sequence (s=2), A329715 (s=3).

Sequence in context: A011108 A331370 A279037 * A154846 A173598 A271727

Adjacent sequences: A328892 A328893 A328894 * A328896 A328897 A328898

Keyword

nonn,cons

Author

Jianing Song, Nov 19 2019

Status

approved