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A328717
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Decimal expansion of Sum_{k>=1} Kronecker(5,k)/k^2.
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7
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7, 0, 6, 2, 1, 1, 4, 0, 3, 2, 5, 9, 7, 4, 0, 9, 6, 9, 9, 3, 1, 0, 0, 3, 1, 7, 5, 7, 6, 2, 5, 6, 4, 0, 2, 7, 6, 6, 0, 2, 4, 6, 4, 7, 1, 8, 5, 2, 9, 4, 6, 8, 6, 3, 9, 4, 2, 1, 1, 7, 4, 0, 2, 1, 6, 5, 6, 7, 7, 6, 0, 4, 4, 3, 8, 3, 8, 3, 0, 0, 7, 6, 8, 3, 3, 7, 4, 5, 6, 6, 4
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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 = A080891 and s = 2.
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LINKS
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H.-J. Seiffert, Problem B-705, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 29, No. 4 (1991), p. 372; An Application of a Series Expansion for (arcsinx)^2, Solution to Problem B-705, ibid., Vol. 31, No. 1 (1993), pp. 85-86.
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FORMULA
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Equals 4*Pi^2/(25*sqrt(5)).
Equals (zeta(2,1/5) - zeta(2,2/5) - zeta(2,3/5) + zeta(2,4/5))/25, where zeta(s,a) is the Hurwitz zeta function.
Equals (polylog(2,u) - polylog(2,u^2) - polylog(2,u^3) + polylog(2,u^4))/sqrt(5), where u = exp(2*Pi*i/5) is a 5th primitive root of unity, i = sqrt(-1).
Equals (polygamma(1,1/5) - polygamma(1,2/5) - polygamma(1,3/5) - polygamma(1,4/5))/25.
Equals Sum_{k>=1} Fibonacci(2*k)/(k^2*binomial(2*k,k)) = Sum_{k>=1} A001906(k)/A002736(k) (Seiffert, 1991). - Amiram Eldar, Jan 17 2022
Equals 1/(Product_{p prime == 1 or 4 (mod 5)} (1 - 1/p^2) * Product_{p prime == 2 or 3 (mod 5)} (1 + 1/p^2)). - Amiram Eldar, Dec 17 2023
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EXAMPLE
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1 - 1/2^2 - 1/3^2 + 1/4^2 + 1/6^2 - 1/7^2 - 1/8^2 + 1/9^2 + ... = 4*Pi^2/(25*sqrt(5)) = 0.70621140325974096993100317576256402766024647185294...
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MATHEMATICA
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RealDigits[4*Pi^2/(25*Sqrt[5]), 10, 102] // First
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PROG
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(PARI) default(realprecision, 100); 4*Pi^2/(25*sqrt(5))
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CROSSREFS
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Decimal expansion of Sum_{k>=1} Kronecker(5,k)/k^s: A086466 (s=1), this sequence (s=2), A328723 (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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