A175573 -id:A175573 - cp's OEIS frontend

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

Hugo Pfoertner, Sep 18 2018

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))."

			0.54321740560665400728765806075511172853510285362260944296039515799...

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.

Cf. A175573, A327996.

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))
```

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

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)

A251992 Decimal expansion of the double infinite sum (negated) sum_{m=1..infinity} sum_{k=0..infinity} (-1)^m/((2k+1)^2+m^2).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Dec 12 2014

			-0.480751144424109780520862631352408574248444731679469...

MathOverflow, How to calculate the infinite sum of this double series?

Cf. A175573.

RealDigits[-Pi*(Pi-Log[2])/16, 10, 103] // First

-Pi*(Pi-log(2))/16.
Also equals sum_{m=1..infinity} (-1)^m*Pi*tanh(m*Pi/2)/(4*m).
Also equals -Pi^2/16 - (Pi/4)*log(theta_2(0,exp(-Pi))) + (Pi/4)*log(theta_3(0,exp(-Pi))), where 'theta' is the elliptic theta function, that is -Pi^2/16 - (Pi/4)*log(A248557) + (Pi/4)*log(A175573).

cp's OEIS Frontend

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

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