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A327135
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Decimal expansion of Sum_{k>=1} Kronecker(-7,k)/k^3.
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6
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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
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OFFSET
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1,3
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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 = A175629 and s = 3.
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LINKS
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FORMULA
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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
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EXAMPLE
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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...
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MATHEMATICA
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RealDigits[32*Pi^3/(343*Sqrt[7]), 10, 102] // First
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PROG
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(PARI) default(realprecision, 100); 32*Pi^3/(343*sqrt(7))
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CROSSREFS
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Decimal expansion of Sum_{k>=1} Kronecker(-7,k)/k^s: A326919 (s=1), A103133 (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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