Linear independence results for the values of divisor functions series (Analytic Number Theory and Related Areas)

Abstract

Let {a_{1}(n)}_{n>1} be a purely periodic sequence of nonnegative integers, not identically zero, and {a_{ell}(n)}_{ngeq 1}overline{(}ell=2, 3, ldots) be the sequences defined inductively by a_{ell}(n):=displaystyle sum_{d|n}a_{ell-1}(d). Then, for an arbitrary integer q(|q|>1), the numbers 1 and displaystyle sum{n=1}^{infty}a{l}(n)q (ell=2, 3, ldots) are linearly independent over mathbb{Q}. In particular, the numbers 1 and displaystyle sum{n=1}^{infty}d{el}(n)q^{-n}(ell=2, 3, . are linearly independent over mathbb{Q}, where d_{ell}(n) are generalized divisor functions.