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A329715
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Decimal expansion of Sum_{k>=1} Kronecker(8,k)/k^3.
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4
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9, 5, 8, 3, 8, 0, 4, 5, 4, 5, 6, 3, 0, 9, 4, 5, 6, 2, 0, 5, 1, 6, 6, 9, 4, 0, 2, 8, 6, 1, 5, 7, 7, 8, 1, 8, 8, 2, 4, 8, 9, 5, 3, 1, 7, 9, 3, 9, 7, 7, 5, 3, 4, 0, 7, 5, 7, 5, 0, 4, 5, 0, 7, 0, 4, 7, 0, 7, 5, 6, 9, 7, 4, 8, 4, 2, 9, 7, 9, 3, 6, 4, 7, 8, 2, 5, 2, 6, 9, 9, 7
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OFFSET
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0,1
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COMMENTS
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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 = 3.
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LINKS
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Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 99.
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FORMULA
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Equals (zeta(3,1/8) - zeta(3,3/8) - zeta(3,5/8) + zeta(3,7/8))/512, where zeta(s,a) is the Hurwitz zeta function.
Equals (polylog(3,u) - polylog(3,u^3) - polylog(3,-u) + polylog(3,-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(2,1/8) - polygamma(2,3/8) - polygamma(2,5/8) + polygamma(2,7/8))/(-1024).
Equals 1/(Product_{p prime == 1 or 7 (mod 8)} (1 - 1/p^3) * Product_{p prime == 3 or 5 (mod 8)} (1 + 1/p^3)). - Amiram Eldar, Dec 17 2023
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EXAMPLE
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1 - 1/3^3 - 1/5^3 + 1/7^3 + 1/9^3 - 1/11^3 - 1/13^3 + 1/15^3 + ... = 0.9583804545...
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MATHEMATICA
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(PolyGamma[2, 1/8] - PolyGamma[2, 3/8] - PolyGamma[2, 5/8] + PolyGamma[2, 7/8])/(-1024) // RealDigits[#, 10, 102] & // First
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CROSSREFS
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Decimal expansion of Sum_{k>=1} Kronecker(8,k)/k^s: A196525 (s=1), A328895 (s=2), this sequence (s=3).
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KEYWORD
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AUTHOR
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STATUS
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approved
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