A346260 -id:A346260 - cp's OEIS frontend

A328895 Decimal expansion of Sum_{k>=1} Kronecker(8,k)/k^2.

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

Offset: 0

Jianing Song, Nov 19 2019

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 = 2.

			1 - 1/3^2 - 1/5^2 + 1/7^2 + 1/9^2 - 1/11^2 - 1/13^2 + 1/15^2 + ... = Pi^2/(8*sqrt(2)) = 0.8723580249...

M. W. Coffey, Summatory relations and prime products for the Stieltjes constants and other related results, arXiv:1701.07064 [math.NT], 2017, eq. (2.1).
R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, Section 2.2 at m=8, r=2, s=2.
Eric Weisstein's World of Mathematics, Dirichlet L-Series.
Eric Weisstein's World of Mathematics, Polygamma Function.

Cf. A091337, A346728, A346727, A346260.
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), A328717 (d=5), this sequence (d=8), A258414 (d=12).
Decimal expansion of Sum_{k>=1} Kronecker(8,k)/k^s: A196525 (s=1), this sequence (s=2), A329715 (s=3).

RealDigits[Pi^2/(8*Sqrt[2]), 10, 102] // First

default(realprecision, 100); Pi^2/(8*sqrt(2))

Equals Pi^2/(8*sqrt(2)).
Equals (zeta(2,1/8) - zeta(2,3/8) - zeta(2,5/8) + zeta(2,7/8))/64, where zeta(s,a) is the Hurwitz zeta function.
Equals (polylog(2,u) - polylog(2,u^3) - polylog(2,-u) + polylog(2,-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(1,1/8) - polygamma(1,3/8) - polygamma(1,5/8) + polygamma(1,7/8))/64.
Equals -Integral_{x=0..oo} log(x)/(x^4 + 1) dx. - Amiram Eldar, Jul 17 2020
Equals 1/(Product_{p prime == 1 or 7 (mod 8)} (1 - 1/p^2) * Product_{p prime == 3 or 5 (mod 8)} (1 + 1/p^2)). - Amiram Eldar, Dec 17 2023

A346728 Decimal expansion of 11 * Pi^4 / (768 * sqrt(2)).

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Jul 30 2021

			0.98654286069397050390153449061672691096683375790950...

L. B. W. Jolley, Summation of Series, Dover, 1961, Eqs. (327), (344).

R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, Section 2.2 at m=8, r=2, s=4.
Michael I. Shamos, Shamos's catalog of the real numbers (2011).
Index entries for transcendental numbers.

Cf. A328895, A346727, A346260, A346451.

RealDigits[11*Pi^4/(768*Sqrt[2]), 10, 120][[1]] (* Amiram Eldar, Jun 13 2023 *)

Equals 11 * Pi^4 / (2^8 * 3 * sqrt(2)).
Equals 1 + Sum_{k>=1} ( (-1)^k/(4*k-1)^4 + (-1)^k/(4*k+1) ).
Equals Sum_{k>=0} (-1)^floor((k+1)/2) / (2*k+1)^4.

A346727 Decimal expansion of 361 * Pi^6 / (245760 * sqrt(2)).

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Jul 30 2021

Shamos (2011) has incorrect formula 19^2*Pi^4 / (2^14*3*5*sqrt(3)).

			0.998573971953530547670270516106668073031954930...

L. B. W. Jolley, Summation of Series, Dover, 1961, Eq. (327).

Michael I. Shamos, Shamos's catalog of the real numbers (2011).
Index entries for transcendental numbers.

Cf. A328895, A346728, A346260, A346450.

RealDigits[361*Pi^6/(245760*Sqrt[2]), 10, 120][[1]] (* Amiram Eldar, Jun 13 2023 *)

Equals 19^2 * Pi^6 / (2^14 * 3 * 5 * sqrt(2)).

Equals 1 + Sum_{k>=1} ( (-1)^k/(4*k-1)^6 + (-1)^k/(4*k+1)^6 ).

cp's OEIS Frontend

A328895 Decimal expansion of Sum_{k>=1} Kronecker(8,k)/k^2.

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A346728 Decimal expansion of 11 * Pi^4 / (768 * sqrt(2)).

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A346727 Decimal expansion of 361 * Pi^6 / (245760 * sqrt(2)).

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