A329716 Decimal expansion of Sum_{k>=1} Kronecker(12,k)/k^3.

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

Offset: 0

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 = A110161 and s = 3.

Examples

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

Links

• Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 99.
• Eric Weisstein's World of Mathematics, Dirichlet L-Series.
• Eric Weisstein's World of Mathematics, Polygamma Function.

Crossrefs

Cf. A110161.

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).

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

Programs

Mathematica

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

Formula

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.

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).

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

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