A086724 Decimal expansion of L(2, chi3) = g(1)-g(2)+g(4)-g(5), where g(k) = Sum_{m>=0} (1/(6*m+k)^2).

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

Offset: 0

Author

N. J. A. Sloane, Jul 31 2003

Keywords

• nonn
• cons

Comments

This number is L(2, chi3), where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3, A102283. - Stuart Clary, Dec 17 2008

Equals 1/1^2 -1/2^2 +1/4^2 -1/5^2 +1/7^2 -1/8^2 +1/10^2 -1/11^2 +-... . This can be split as (1/1^2 -1/5^2 +1/7^2 -1/11^2 +-...) - (1/2^2 -1/4^2 +1/8^2 -1/10^2..) = (g(1)-g(5)) - (g(2)-g(4)). The first of these two series is A214552 and the second series is 1/(2^2)*(1-1/2^2 +1/4^2-1/5^2+-...), namely a quarter of the original series. Therefore 5/4 of this value here equals A214552. - R. J. Mathar, Jul 20 2012

Calegari, Dimitrov, & Tang prove that this number is irrational. - Charles R Greathouse IV, Aug 29 2024

Examples

			0.781302412896486296867...
		

References

• L. Fejes Toth, Lagerungen in der Ebene, auf der Kugel und im Raum, 2nd. ed., Springer-Verlag, Berlin, Heidelberg 1972; see p. 213.

Links

• D. H. Bailey, J. M. Borwein, and R. E. Crandall, Integrals of the Ising class, J. Phys. A 39 (2006) 12271, variable C_3.
• Frank Calegari, Vesselin Dimitrov, and Yunqing Tang, The linear independence of 1, ζ(2), and L(2,χ₋₃), arXiv:2408.15403 [math.NT], 2024.
• Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 98.
• Richard J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015, L(m=3,r=2,s=2).
• Lorenz Milla, World record computation of Dirichlet L(-3,2) series (1,000,000,000,000 digits)
• Kh. Hessami Pilehrood and T. Hessami Pilehrood, Bivariate identities for values of the Hurwitz zeta function and supercongruences, El. J. Combin. 18 (2) (2012), Article P35, value of K after Theorem 4.

Crossrefs

Cf. A086722-A086731, A102283, A214549 (principal character), A214552.

Cf. A153066, A153067, A153068.

Programs

Mathematica

nmax = 1000; First[ RealDigits[(Zeta[2, 1/3] - Zeta[2, 2/3])/9, 10, nmax] ] (* Stuart Clary, Dec 17 2008 *)

PARI

zetahurwitz(2,1/3)/9 - zetahurwitz(2,2/3)/9 \\ Charles R Greathouse IV, Jan 30 2018

Formula

From Jean-François Alcover, Jul 17 2014, updated Jan 23 2015: (Start)

Equals Sum_{n>=1} jacobi(-3, n+3)/n^2.

Equals (8/15)*4F3(1/2,1,1,2; 5/4,3/2,7/4; 3/4), where 4F3 is the generalized hypergeometric function.

Equals 4*Pi*log(3)/(3*sqrt(3)) - 4*Integral_{0..1} log(x+1)/(x^2-x+1) dx. (End)

Equals Product_{p prime} (1 - Kronecker(-3, p)/p^2)^(-1) = Product_{p prime != 3} (1 + (-1)^(p mod 3)/p^2)^(-1). - Amiram Eldar, Nov 06 2023