A103133 -id:A103133 - 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

A328717 Decimal expansion of Sum_{k>=1} Kronecker(5,k)/k^2.

Original entry on oeis.org

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

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 = A080891 and s = 2.

			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...

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).
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.
Eric Weisstein's World of Mathematics, Dirichlet L-Series.
Eric Weisstein's World of Mathematics, Polygamma Function.

Cf. A000045, A001906, A002736, A080891.
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).
Decimal expansion of Sum_{k>=1} Kronecker(5,k)/k^s: A086466 (s=1), this sequence (s=2), A328723 (s=3).

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

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

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

A327135 Decimal expansion of Sum_{k>=1} Kronecker(-7,k)/k^3.

Original entry on oeis.org

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

Offset: 1

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 = A175629 and s = 3.

			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...

R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015, L(m=7,r=4,s=3).
Eric Weisstein's World of Mathematics, Dirichlet L-Series.
Eric Weisstein's World of Mathematics, Polygamma Function.

Cf. A175629.
Decimal expansion of Sum_{k>=1} Kronecker(d,k)/k^3, where d is a fundamental discriminant: A251809 (d=-8), this sequence (d=-7), A153071 (d=-4), A129404 (d=-3), A002117 (d=1), A328723 (d=5), A329715 (d=8), A329716 (d=12).
Decimal expansion of Sum_{k>=1} Kronecker(-7,k)/k^s: A326919 (s=1), A103133 (s=2), this sequence (s=3).

RealDigits[32*Pi^3/(343*Sqrt[7]), 10, 102] // First

default(realprecision, 100); 32*Pi^3/(343*sqrt(7))

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

A309710 Decimal expansion of Sum_{k>=1} Kronecker(-8,k)/k^2.

Original entry on oeis.org

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

Offset: 1

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 = A188510 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 + ...= 1.0647341710...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 98.
R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, Section 2.2 L(m=8, r=4, s=2).
Eric Weisstein's World of Mathematics, Dirichlet L-Series.
Eric Weisstein's World of Mathematics, Polygamma Function.

Cf. A188510.
Decimal expansion of Sum_{k>=1} Kronecker(d,k)/k^2, where d is a fundamental discriminant: this sequence (d=-8), A103133 (d=-7), A006752 (d=-4), A086724 (d=-3), A013661 (d=1), A328717 (d=5), A328895 (d=8), A258414 (d=12).
Decimal expansion of Sum_{k>=1} Kronecker(-8,k)/k^s: A093954 (s=1), this sequence (s=2), A251809 (s=3).

(PolyGamma[1, 1/8] + PolyGamma[1, 3/8] - PolyGamma[1, 5/8] - PolyGamma[1, 7/8])/64 // RealDigits[#, 10, 102] & // First

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 1/(Product_{p prime == 1 or 3 (mod 8)} (1 - 1/p^2) * Product_{p prime == 5 or 7 (mod 8)} (1 + 1/p^2)). - Amiram Eldar, Dec 17 2023

A326919 Decimal expansion of Sum_{k>=1} Kronecker(-7,k)/k.

Original entry on oeis.org

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

Offset: 1

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 = A175629 and s = 1.

			1 + 1/2 - 1/3 + 1/4 - 1/5 - 1/6 + 1/8 + 1/9 - 1/10 + 1/11 - 1/12 - 1/13 + ... = Pi/sqrt(7) = 1.1874104117...

Sergio A. Carrillo, Where did the examples of Abel's continuity theorem go?, arXiv:2010.10290 [math.HO], 2020. See p. 11.
Eric Weisstein's World of Mathematics, Class Number.
Eric Weisstein's World of Mathematics, Dirichlet L-Series.
Eric Weisstein's World of Mathematics, Polygamma Function.

Cf. A175629.
Decimal expansion of Sum_{k>=1} Kronecker(d,k)/k, where d is a fundamental discriminant: A093954 (d=-8), this sequence (d=-7), A003881 (d=-4), A073010 (d=-3), A086466 (d=5), A196525 (d=8), A196530 (d=12).
Decimal expansion of Sum_{k>=1} Kronecker(-7,k)/k^s: this sequence (s=1), A103133 (s=2), A327135 (s=3).

RealDigits[Pi/Sqrt[7], 10, 102] // First

PARI
```
default(realprecision, 100); Pi/sqrt(7)
```

Equals Pi/sqrt(7). This is related to the class number formula: if d<0 is the fundamental discriminant of an imaginary quadratic number field, Chi(k) = Kronecker(d,k), then L(1,Chi) = Sum_{k>=1} Kronecker(d,k)/k = 2*Pi*h(d)/(sqrt(|d|)*w(d)), where h(d) is the class number of K = Q[sqrt(d)], w(d) is the number of elements in K whose norms are 1 (w(d) = 6 if d = -3, 4 if d = -4 and 2 if d < -4). Here d = -7, h(d) = 1, w(d) = 2.
Equals (polylog(1,u) + polylog(1,u^2) - polylog(1,u^3) + polylog(1,u^4) - polylog(1,u^5) - polylog(1,u^6))/sqrt(-7), where u = exp(2*Pi*i/7) is a 7th primitive root of unity, i = sqrt(-1).
Equals (polygamma(0,1/7) + polygamma(0,2/7) - polygamma(0,3/7) + polygamma(0,4/7) - polygamma(0,5/7) - polygamma(0,6/7))/49.
Equals 1/Product_{p prime} (1 - Kronecker(-7,p)/p), where Kronecker(-7,p) = 0 if p = 7, 1 if p == 1, 2 or 4 (mod 7) or -1 if p == 3, 5 or 6 (mod 7). - Amiram Eldar, Dec 17 2023

cp's OEIS Frontend

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

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A328717 Decimal expansion of Sum_{k>=1} Kronecker(5,k)/k^2.

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A327135 Decimal expansion of Sum_{k>=1} Kronecker(-7,k)/k^3.

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A309710 Decimal expansion of Sum_{k>=1} Kronecker(-8,k)/k^2.

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A326919 Decimal expansion of Sum_{k>=1} Kronecker(-7,k)/k.

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