# Multidimensional series: an application of quantum field theory

While computing the quantum vacuum energy of a real scalar field defined on $$mathbb{R}times mathbb{T}^3$$, I encountered the following sum:

$$sum_{n_1^2+n_2^2+n_3^2geq 1}^{infty} frac{1}{(n_1^2+n_2^2+n_3^2)^2}$$

Does it have a known exact result?

Thank you.

EDIT: What is the value of Epstein $$zeta_3(2)$$? Where the Epstein zeta function is defined as
$$zeta_k(s)=sum_{mathbb{Z}^k/{0}} frac{1}{(n_1^2+….n_k^2)^s}$$

MathOverflow Asked by Andrea T on November 22, 2021

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