A328723 Decimal expansion of Sum_{k>=1} Kronecker(5,k)/k^3.

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

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 = A080891 and s = 3.

Examples

			1 - 1/2^3 - 1/3^3 + 1/4^3 + 1/6^3 - 1/7^3 - 1/8^3 + 1/9^3 + ... = 0.8548247666...
		

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. A080891.

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), this sequence (d=5), A329715 (d=8), A329716 (d=12).

Decimal expansion of Sum_{k>=1} Kronecker(5,k)/k^s: A086466 (s=1), A328717 (s=2), this sequence (s=3).

Programs

Mathematica

(PolyGamma[2, 1/5] - PolyGamma[2, 2/5] - PolyGamma[2, 3/5] + PolyGamma[2, 4/5])/(-250) // RealDigits[#, 10, 102] & // First

Formula

Equals (zeta(3,1/5) - zeta(3,2/5) - zeta(3,3/5) + zeta(3,4/5))/25, where zeta(s,a) is the Hurwitz zeta function.

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

Equals (polygamma(2,1/5) - polygamma(2,2/5) - polygamma(2,3/5) - polygamma(2,4/5))/(-250).

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