Linear independence of certain numbers

Abstract

Let $k\ge 2$, $b\ge 2$ and $1\le a_1<a_2<\cdots <a_m$ be integers such that $\root k \of {a_i/a_j}\notin \mathbb {Q}$ for any $i\ne j$. In this article, we prove that the real numbers

$$\begin{aligned} 1,\quad \sum _{n=1}^{\infty }\frac{1}{b^{a_1 n^k}}, \quad \sum _{n=1}^{\infty }\frac{1}{b^{a_2 n^k}},\ldots , \sum _{n=1}^{\infty }\frac{1}{b^{a_m n^k}} \end{aligned}$$

are $\mathbb {Q}$-linearly independent.