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Decimal expansion of Sum_{k>=1} Kronecker(5,k)/k^2.
7

%I #63 Dec 17 2023 03:06:24

%S 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,

%T 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,

%U 7,6,0,4,4,3,8,3,8,3,0,0,7,6,8,3,3,7,4,5,6,6,4

%N Decimal expansion of Sum_{k>=1} Kronecker(5,k)/k^2.

%C 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).

%C 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.

%C 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.

%C 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)!).

%C In this sequence we have Chi = A080891 and s = 2.

%H M. W. Coffey, <a href="https://arxiv.org/abs/1701.07064">Summatory relations and prime products for the Stieltjes constants and other related results</a>, arXiv:1701.07064 [math.NT], 2017, eq. (2.1).

%H H.-J. Seiffert, <a href="https://fq.math.ca/Scanned/29-4/elementary29-4.pdf">Problem B-705</a>, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 29, No. 4 (1991), p. 372; <a href="https://www.fq.math.ca/Scanned/31-1/elementary31-1.pdf">An Application of a Series Expansion for (arcsinx)^2</a>, Solution to Problem B-705, ibid., Vol. 31, No. 1 (1993), pp. 85-86.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DirichletL-Series.html">Dirichlet L-Series</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PolygammaFunction.html">Polygamma Function</a>.

%F Equals 4*Pi^2/(25*sqrt(5)).

%F 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.

%F 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).

%F Equals (polygamma(1,1/5) - polygamma(1,2/5) - polygamma(1,3/5) - polygamma(1,4/5))/25.

%F 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

%F 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

%e 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...

%t RealDigits[4*Pi^2/(25*Sqrt[5]), 10, 102] // First

%o (PARI) default(realprecision, 100); 4*Pi^2/(25*sqrt(5))

%Y Cf. A000045, A001906, A002736, A080891.

%Y Decimal expansion of Sum_{k>=1} Kronecker(d,k)/k^2, where d is a fundamental discriminant: A309710 (d=-8), A103133 (d=-7), A006752 (d=-4), A086724 (d=-3), A013661 (d=1), this sequence (d=5), A328895 (d=8), A258414 (d=12).

%Y Decimal expansion of Sum_{k>=1} Kronecker(5,k)/k^s: A086466 (s=1), this sequence (s=2), A328723 (s=3).

%K nonn,cons

%O 0,1

%A _Jianing Song_, Nov 19 2019