A327135 Decimal expansion of Sum_{k>=1} Kronecker(-7,k)/k^3.

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

Offset: 1

Author

Jianing Song, Nov 19 2019

Keywords

• nonn
• cons

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 = A175629 and s = 3.

Examples

			1 + 1/2^3 - 1/3^3 + 1/4^3 - 1/5^3 - 1/6^3 + 1/8^3 + 1/9^3 - 1/10^3 + 1/11^3 - 1/12^3 - 1/13^3 + ... = 32*Pi^3/(343*sqrt(7)) = 1.0933430694...
		

Links

• R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015, L(m=7,r=4,s=3).
• Eric Weisstein's World of Mathematics, Dirichlet L-Series.
• Eric Weisstein's World of Mathematics, Polygamma Function.

Crossrefs

Cf. A175629.

Decimal expansion of Sum_{k>=1} Kronecker(d,k)/k^3, where d is a fundamental discriminant: A251809 (d=-8), this sequence (d=-7), A153071 (d=-4), A129404 (d=-3), A002117 (d=1), A328723 (d=5), A329715 (d=8), A329716 (d=12).

Decimal expansion of Sum_{k>=1} Kronecker(-7,k)/k^s: A326919 (s=1), A103133 (s=2), this sequence (s=3).

Programs

Mathematica

RealDigits[32*Pi^3/(343*Sqrt[7]), 10, 102] // First

PARI

default(realprecision, 100); 32*Pi^3/(343*sqrt(7))

Formula

Equals 32*Pi^3/(343*sqrt(7)).

Equals (zeta(3,1/7) + zeta(3,2/7) - zeta(3,3/7) + zeta(3,4/7) - zeta(3,5/7) - zeta(3,6/7))/343.

Equals (polylog(3,u) + polylog(3,u^2) - polylog(3,u^3) + polylog(3,u^4) - polylog(3,u^5) - polylog(3,u^6))/sqrt(-7), where u = exp(2*Pi*i/7) is a 7th primitive root of unity, i = sqrt(-1).

Equals (polygamma(2,1/7) + polygamma(2,2/7) - polygamma(2,3/7) + polygamma(2,4/7) - polygamma(2,5/7) - polygamma(2,6/7))/(-686).

Equals 1/(Product_{p prime == 1, 2 or 4 (mod 7)} (1 - 1/p^3) * Product_{p prime == 3, 5 or 6 (mod 7)} (1 + 1/p^3)). - Amiram Eldar, Dec 17 2023