A086466 -id:A086466 - cp's OEIS frontend

A328723 Decimal expansion of Sum_{k>=1} Kronecker(5,k)/k^3.

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

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 = 3.

			1 - 1/2^3 - 1/3^3 + 1/4^3 + 1/6^3 - 1/7^3 - 1/8^3 + 1/9^3 + ... = 0.8548247666...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 99.
Eric Weisstein's World of Mathematics, Dirichlet L-Series.
Eric Weisstein's World of Mathematics, Polygamma Function.

Cf. A080891.
Decimal expansion of Sum_{k>=1} Kronecker(d,k)/k^3, where d is a fundamental discriminant: A251809 (d=-8), A327135 (d=-7), A153071 (d=-4), A129404 (d=-3), A002117 (d=1), this sequence (d=5), A329715 (d=8), A329716 (d=12).
Decimal expansion of Sum_{k>=1} Kronecker(5,k)/k^s: A086466 (s=1), A328717 (s=2), this sequence (s=3).

(PolyGamma[2, 1/5] - PolyGamma[2, 2/5] - PolyGamma[2, 3/5] + PolyGamma[2, 4/5])/(-250) // RealDigits[#, 10, 102] & // First

Equals (zeta(3,1/5) - zeta(3,2/5) - zeta(3,3/5) + zeta(3,4/5))/25, where zeta(s,a) is the Hurwitz zeta function.
Equals (polylog(3,u) - polylog(3,u^2) - polylog(3,u^3) + polylog(3,u^4))/sqrt(5), where u = exp(2*Pi*i/5) is a 5th primitive root of unity, i = sqrt(-1).
Equals (polygamma(2,1/5) - polygamma(2,2/5) - polygamma(2,3/5) - polygamma(2,4/5))/(-250).
Equals 1/(Product_{p prime == 1 or 4 (mod 5)} (1 - 1/p^3) * Product_{p prime == 2 or 3 (mod 5)} (1 + 1/p^3)). - Amiram Eldar, Dec 17 2023

A241519 Denominators of b(n) = b(n-1)/2 + 1/(2*n), b(0)=0.

Original entry on oeis.org

1, 2, 2, 12, 3, 15, 60, 840, 105, 630, 630, 13860, 6930, 180180, 360360, 144144, 9009, 306306, 306306, 11639628, 14549535, 14549535, 58198140, 2677114440, 334639305, 3346393050

Offset: 0

Paul Curtz, Apr 24 2014

nonn

Generally, 2*b(n) = b(n-1) + f(n). See, for f(n)=n, A000337(n)/2^n.
a(0)=1. b(n) is mentioned in A241269.
Difference table of b(n):
     0,   1/2,   1/2,  5/12,    1/3,   4/15, ...
   1/2,     0, -1/12, -1/12,  -1/15,  -1/20, ...
  -1/2, -1/12,     0,  1/60,   1/60, 11/840, ...
  5/12,  1/12,  1/60,     0, -1/280, -1/280, ...
etc.
b(n) is mentioned in A241269 as an autosequence of the first kind.
The denominators of the first two upper diagonals are the positive Apéry numbers, A005430(n+1). Compare to the array in A003506.
Numerators: 0, 1, 1, 5, 1, 4, 13, 151, 16, 83, 73, 1433, 647, 15341, ... .

			0, 1/2, 1/2, 5/12, 1/3, 4/15, 13/60, 151/840, 16/105, 83/630, 73/630, ...
b(1) = (0+1)/2, hence a(1)=2.
b(2) = (1/2+1/2)/2 = 1/2, hence a(2)=2.
b(3) = (1/2+1/3)/2 = 5/12, hence a(3)=12.

Cf. A086466.

Cf. A242376 (numerators).

b[0] = 0; b[n_] := b[n] = 1/2*(b[n-1] + 1/n); Table[b[n] // Denominator, {n, 0, 25}] (* Jean-François Alcover, Apr 25 2014 *)
Table[-Re[LerchPhi[2, 1, n + 1]], {n, 0, 20}] // Denominator (* Eric W. Weisstein, Dec 11 2017 *)
-Re[LerchPhi[2, 1, Range[20]]] // Denominator (* Eric W. Weisstein, Dec 11 2017 *)
RecurrenceTable[{b[n] == b[n - 1]/2 + 1/(2 n), b[0] == 0}, b[n], {n, 20}] // Denominator (* Eric W. Weisstein, Dec 11 2017 *)

b(n) = -Re(Phi(2, 1, n + 1)) where Phi denotes the Lerch transcendent. - Eric W. Weisstein, Dec 11 2017

Extension, after a(13), from Jean-François Alcover, Apr 24 2014

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

A145435 Decimal expansion of log(1/2 + 1/sqrt(2))/sqrt(5).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Feb 08 2009

This is an erroneous version of A086466 produced by the Apelblat formula, which contains two typos.

			0.084177408000833203035548695384667267885531840399884582887759..

Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, (1996), 4.1.42.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers

Cf. A014176, A020762.

SetDefaultRealField(RealField(100)); 1/5*Log(1/2+1/2*2^(1/2))*5^(1/2); // G. C. Greubel, Oct 02 2018

Maple
```
1/5*ln(1/2+1/2*2^(1/2))*5^(1/2) ;
```

Join[{0},RealDigits[Log[1/2+1/Sqrt[2] ]/Sqrt[5],10,120][[1]]] (* Harvey P. Dale, May 24 2016 *)

default(realprecision, 100); 1/5*log(1/2+1/2*2^(1/2))*5^(1/2) \\ G. C. Greubel, Oct 02 2018

Equals log(A014176/2)*A020762.

Uncovered Apelblat errors. - R. J. Mathar, Mar 04 2009

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A328723 Decimal expansion of Sum_{k>=1} Kronecker(5,k)/k^3.

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A241519 Denominators of b(n) = b(n-1)/2 + 1/(2*n), b(0)=0.

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

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A145435 Decimal expansion of log(1/2 + 1/sqrt(2))/sqrt(5).

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