In this blog we prove a Ramanujan-type identity:

$$$\sum\limits_{n_1 \in \mathbb{Z}}(-1)^{n_1}\sum\limits_{(n_2, n_3) \in \mathbb{Z}^2}\frac{1}{\sqrt{n_1^2+ (n_2+0.5)^2+(n_3+0.5)^2}\sinh{(\pi\sqrt{n_1^2+ (n_2+0.5)^2+(n_3+0.5)^2})}} = 1$$$. (1)

from math import *

ans = 0
for m1 in range(-100, 101):
    ans1 = 0
    for m2 in range(-100, 101):
        for m3 in range(-100, 101):
            p = m1**2 + (m2+0.5)**2 + (m3+0.5)**2
            q = sqrt(p)
            r = sinh(pi*q)
            ans1 += 1/(q*r)
    print(ans1, m1)
    ans += (1 if m1%2==0 else -1) * ans1
print(ans)

ans = 0
for m1 in range(-100, 101):
    ans1 = 0
    for m2 in range(-100, 101):
        p = m1**2 + (m2+0.5)**2
        q = sqrt(p)
        r = sinh(pi*q)
        ans1 += 1/(q*r)
    #print(ans1, m1)
    ans += (1 if m1%2==0 else -1) * ans1
print(ans)

First, we consider a 4D lattice sum:

$$$S(n_1, n_2, n_3, n_4) := \sum\limits_{(n_1, n_2, n_3, n_4) \in \mathbb{Z}^4} \frac{(-1)^{n_1+n_4}}{n_1^2 + (n_2+0.5)^2 + (n_3+0.5)^2 + n_4^2}$$$. We will show later that $$$S(n_1, n_2, n_3, n_4)$$$ equals to $$$\pi$$$.

For $$$\lambda > 0$$$, $$$\int_{0}^\infty e^{-\lambda x}dx = \frac{1}{\lambda}$$$, therefore $$$S(n_1, n_2, n_3, n_4) = \sum\limits_{(n_1, n_2, n_3, n_4) \in \mathbb{Z}^4} \int_{0}^\infty (-1)^{n_1+n_4} \exp((-n_1^2-(n_2+0.5)^2-(n_3+0.5)^2-n_4^2)x)dx$$$.

By the Poisson summation formula of the theta function, i.e.,

$$$\sum\limits_{n \in \mathbb{Z}}(-1)^n \exp(-n^2 x) = \sqrt{\frac{\pi}{x}}\exp(\frac{-(n_4+0.5)^2\pi^2}{x})$$$