Linear independence of certain numbers (Q1737358)

The paper deals with linear independence of infinite series over the rational numbers. The main result states the following. Let \(k\ge2\), \(b\ge2\) and \(1\le a_1 < a_2 < \dots < a_m\) be integers such that \(\sqrt[k]{\frac{a_i}{a_j}}\not\in\mathbb Q\) for any \(i\ne j\). Then the real numbers \[ 1,\quad \sum_{n=1}^\infty \frac{1}{b^{a_1n^k}} , \quad \sum_{n=1}^\infty \frac{1}{b^{a_2n^k}},\, \quad \ldots,\quad \sum_{n=1}^\infty \frac{1}{b^{a_mn^k}} \] are linearly independent over the rational numbers.