Linear independence of certain Lambert series

Abstract:

We prove that if $q\ne 0,\pm 1$ and $\ell \ge 1$ are fixed integers, then the numbers \[ 1,\quad \sum _{n= 1} \frac {1}{q^{n}-1},\quad \sum _{n=1}^{\infty }\frac {1}{q^{n^2}-1},\quad \dots ,\quad \sum _{n=1}^{\infty }\frac {1}{q^{n^{\ell }}-1} \] are linearly independent over ${\mathbb Q}$. This generalizes a result of Erdős, who treated the case $\ell =1$. The method is based on the original approaches of Chowla and Erdős, together with some results about primes in arithmetic progressions with large moduli of Ahlford, Granville and Pomerance.