A153071 Decimal expansion of L(3, chi4), where L(s, chi4) is the Dirichlet L-function for the non-principal character modulo 4.

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

Offset: 0

Author

Stuart Clary, Dec 17 2008

Keywords

• nonn
• cons
• easy

Examples

			L(3, chi4) = Pi^3/32 = 0.9689461462593693804836348458469186...
		

References

• Bruce C. Berndt, Ramanujan's Notebooks, Part II, Springer-Verlag, 1989. See page 293, Entry 25 (iii).
• Leonhard Euler, Introductio in Analysin Infinitorum, First Part, Articles 175, 284 and 287.
• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.4.1, p. 20.

Links

• J. T. Groenman, Problem 1511, Crux Mathematicorum, Vol. 16, No. 2 (1990), p. 43; Solution to Problem 1511, by Beatriz Margolis, ibid., Vol. 17, No. 3 (1991), pp. 92-93.
• Qing-Hu Hou and Zhi-Wei Sun, A q-analogue of the identity Sum_{k>=0}(-1)^k/(2k+1)^3 = Pi^3/32, arXiv:1808.04717 [math.CO], 2018.
• Masato Kobayashi, Integral representations for zeta(3) with the inverse sine function, arXiv:2108.01247 [math.NT], 2021.
• Richard J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015, section 2.2 entry L(m=4,r=2,s=3).
• Michael Penn, An infinite tangent product, YouTube video, 2020.
• Eric Weisstein's World of Mathematics, Dirichlet Beta Function.
• Wikipedia, Dirichlet beta function.
• Index entries for transcendental numbers.

Crossrefs

Cf. A101455, A153072, A153073, A153074.

Cf. A233091, A251809, A331095.

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)).

Programs

Mathematica

nmax = 1000; First[ RealDigits[Pi^3/32, 10, nmax] ]

PARI

Pi^3/32 \\ Michel Marcus, Aug 15 2018

Formula

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.

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 + ...

Series: L(3, chi4) = Sum_{k>=0} tanh((2k+1) Pi/2)/(2k+1)^3. [Ramanujan; see Berndt, page 293]

Closed form: L(3, chi4) = Pi^3/32 = 1/A331095.

Equals Sum_{n>=0} (-1)^n/(2*n+1)^3. - Jean-François Alcover, Mar 29 2013

Equals Product_{k>=3} (1 - tan(Pi/2^k)^4) (Groenman, 1990). - Amiram Eldar, Apr 03 2022

Equals Integral_{x=0..1} arcsinh(x)*arccos(x)/x dx (Kobayashi, 2021). - Amiram Eldar, Jun 23 2023

From Amiram Eldar, Nov 06 2023: (Start)

Equals beta(3), where beta is the Dirichlet beta function.

Equals Product_{p prime >= 3} (1 - (-1)^((p-1)/2)/p^3)^(-1). (End)

Extensions

Offset corrected by R. J. Mathar, Feb 05 2009