A086724 - OEIS

%I #64 Aug 29 2024 15:03:17

%S 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,

%T 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,

%U 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

%N 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).

%C 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

%C 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

%C Calegari, Dimitrov, & Tang prove that this number is irrational. - _Charles R Greathouse IV_, Aug 29 2024

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

%H D. H. Bailey, J. M. Borwein, and R. E. Crandall, <a href="http://dx.doi.org/10.1088/0305-4470/39/40/001">Integrals of the Ising class</a>, J. Phys. A 39 (2006) 12271, variable C_3.

%H Frank Calegari, Vesselin Dimitrov, and Yunqing Tang, <a href="https://arxiv.org/abs/2408.15403">The linear independence of 1, ζ(2), and L(2,χ₋₃)</a>, arXiv:2408.15403 [math.NT], 2024.

%H Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 98.

%H Richard J. Mathar, <a href="https://arxiv.org/abs/1008.2547">Table of Dirichlet L-series and prime zeta modulo functions for small moduli</a>, arXiv:1008.2547 [math.NT], 2010-2015, L(m=3,r=2,s=2).

%H Kh. Hessami Pilehrood and T. Hessami Pilehrood, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v18i2p35">Bivariate identities for values of the Hurwitz zeta function and supercongruences</a>, El. J. Combin. 18 (2) (2012), Article P35, value of K after Theorem 4.

%F From _Jean-François Alcover_, Jul 17 2014, updated Jan 23 2015: (Start)

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

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

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

%F 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

%e 0.781302412896486296867...

%t nmax = 1000; First[ RealDigits[(Zeta[2, 1/3] - Zeta[2, 2/3])/9, 10, nmax] ] (* _Stuart Clary_, Dec 17 2008 *)

%o (PARI) zetahurwitz(2,1/3)/9 - zetahurwitz(2,2/3)/9 \\ _Charles R Greathouse IV_, Jan 30 2018

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

%Y Cf. A153066, A153067, A153068.

%K nonn,cons

%O 0,1

%A _N. J. A. Sloane_, Jul 31 2003