Let \(m \ge 2\) be any integer and let \(\beta >1\) be a real algebraic integer such that all its other Galois conjugates have absolute value less than or equal to 1. Let \(a_1, a_2, \ldots , a_m\) be distinct positive integers. In this article we prove that the following infinite sums \(\begin{aligned} 1, \quad \sum _{n=1}^{\infty }\frac{1}{\beta ^{a_1 n^2}}, \quad \sum _{n=1}^{\infty }\frac{1}{\beta ^{a_2 n^2}}, \ldots , \sum _{n=1}^{\infty }\frac{1}{\beta ^{a_m n^2}} \end{aligned}\) are \(\mathbb {Q}(\beta )\) -linearly independent. As a consequence, we prove the linear independence of special values of Jacobi-theta constants.
