A319332 Decimal expansion of 1/2 + Sum_{n>0} exp(-Pi*n^2).

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

Offset: 0

Author

Hugo Pfoertner, Sep 18 2018

Keywords

• nonn
• cons

Comments

A part of Ramanujan's question 629 in the Journal of the Indian Mathematical Society (VII, 40) asked "... deduce the following: 1/2 + Sum_{n>=1} exp(-Pi*n^2) = sqrt(5*sqrt(5)-10) * (1/2 + Sum_{n>=1} exp(-5*Pi*n^2))."

Examples

			0.54321740560665400728765806075511172853510285362260944296039515799...
		

Links

• B. C. Berndt, Y. S. Choi, and S. Y. Kang, The problems submitted by Ramanujan to the Journal of Indian Math. Soc., in: Continued fractions, Contemporary Math., 236 (1999), 15-56, DOI: 10.1090/conm/236 (see Q629, JIMS VII).
• B. C. Berndt, Y. S. Choi, and S. Y. Kang, The problems submitted by Ramanujan to the Journal of Indian Math. Soc., in: Continued fractions, Contemporary Math., 236 (1999), 15-56 (see Q629, JIMS VII).
• Dan Romik, The Taylor coefficients of the Jacobi theta_3, arXiv:1807.06130 [math.NT], 2018.

Crossrefs

Cf. A175573, A327996.

Programs

Mathematica

RealDigits[Pi^(1/4)/(2*Gamma[3/4]), 10, 120][[1]] (* Amiram Eldar, May 30 2023 *)

PARI

1/2+suminf(n=1,exp(-Pi*n*n))

PARI

sqrt(5*sqrt(5)-10)*(1/2+suminf(n=1,exp(-5*Pi*n*n)))

Formula

Equals Pi^(1/4)/(2*Gamma(3/4)). - Peter Luschny, Jun 11 2020

From Amiram Eldar, May 30 2023: (Start)

Equals Gamma(1/4)/(2*sqrt(2)*Pi^(3/4)).

Equals A327996 / sqrt(Pi). (End)