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%I #41 Nov 07 2023 07:25:31
%S 9,6,8,9,4,6,1,4,6,2,5,9,3,6,9,3,8,0,4,8,3,6,3,4,8,4,5,8,4,6,9,1,8,6,
%T 0,0,0,6,9,5,4,0,2,6,7,6,8,3,9,0,9,6,1,5,4,4,2,0,1,6,8,1,5,7,4,3,9,4,
%U 9,8,4,1,1,7,0,8,0,3,3,1,3,6,7,3,9,5,9,4,0,7
%N Decimal expansion of L(3, chi4), where L(s, chi4) is the Dirichlet L-function for the non-principal character modulo 4.
%D Leonhard Euler, Introductio in Analysin Infinitorum, First Part, Articles 175, 284 and 287.
%D Bruce C. Berndt, Ramanujan's Notebooks, Part II, Springer-Verlag, 1989. See page 293, Entry 25 (iii).
%H J. T. Groenman, <a href="https://cms.math.ca/publications/crux/issue?volume=16&issue=2">Problem 1511</a>, Crux Mathematicorum, Vol. 16, No. 2 (1990), p. 43; <a href="https://cms.math.ca/publications/crux/issue?volume=17&issue=3">Solution to Problem 1511</a>, by Beatriz Margolis, ibid., Vol. 17, No. 3 (1991), pp. 92-93.
%H Qing-Hu Hou and Zhi-Wei Sun, <a href="https://arxiv.org/abs/1808.04717">A q-analogue of the identity Sum_{k>=0}(-1)^k/(2k+1)^3 = Pi^3/32</a>, arXiv:1808.04717 [math.CO], 2018.
%H Masato Kobayashi, <a href="https://arxiv.org/abs/2108.01247">Integral representations for zeta(3) with the inverse sine function</a>, arXiv:2108.01247 [math.NT], 2021.
%H Richard J. Mathar, <a href="http://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, section 2.2 entry L(m=4,r=2,s=3).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DirichletBetaFunction.html">Dirichlet Beta Function</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Dirichlet_beta_function">Dirichlet beta function</a>.
%F chi4(k) = Kronecker(-4, k); chi4(k) is 0, 1, 0, -1 when k reduced modulo 4 is 0, 1, 2, 3, respectively; chi4 is A101455.
%F Series: L(3, chi4) = Sum_{k>=1} chi4(k) k^{-3} = 1 - 1/3^3 + 1/5^3 - 1/7^3 + 1/9^3 - 1/11^3 + 1/13^3 - 1/15^3 + ...
%F Series: L(3, chi4) = Sum_{k>=0} tanh((2k+1) Pi/2)/(2k+1)^3. [Ramanujan; see Berndt, page 293]
%F Closed form: L(3, chi4) = Pi^3/32.
%F Equals Sum_{n>=0} (-1)^n/(2*n+1)^3. - _Jean-François Alcover_, Mar 29 2013
%F Equals Product_{k>=3} (1 - tan(Pi/2^k)^4) (Groenman, 1990). - _Amiram Eldar_, Apr 03 2022
%F Equals Integral_{x=0..1} arcsinh(x)*arccos(x)/x dx (Kobayashi, 2021). - _Amiram Eldar_, Jun 23 2023
%F From _Amiram Eldar_, Nov 06 2023: (Start)
%F Equals beta(3), where beta is the Dirichlet beta function.
%F Equals Product_{p prime >= 3} (1 - (-1)^((p-1)/2)/p^3)^(-1). (End)
%e L(3, chi4) = Pi^3/32 = 0.9689461462593693804836348458469186...
%t nmax = 1000; First[ RealDigits[Pi^3/32, 10, nmax] ]
%o (PARI) Pi^3/32 \\ _Michel Marcus_, Aug 15 2018
%Y Cf. A101455, A153072, A153073, A153074.
%Y Cf. A233091, A251809. [_Bruno Berselli_, Dec 10 2014]
%Y Cf. A003881 (beta(1)=Pi/4), A006752 (beta(2)=Catalan), A175572 (beta(4)), A175571 (beta(5)), A175570 (beta(6)), A258814 (beta(7)), A258815 (beta(8)), A258816 (beta(9)).
%K nonn,cons,easy
%O 0,1
%A _Stuart Clary_, Dec 17 2008
%E Offset corrected by _R. J. Mathar_, Feb 05 2009
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