Refinement of the Chowla–Erdős method and linear independence of certain Lambert series

In this paper, we refine the method of Chowla and Erdős on the irrationality of Lambert series and study a necessary condition for the infinite series ∑ θ ⁢ ( n ) / q n {\sum\theta(n)/q^{n}} to be a rational number, where q is an integer with | q | > 1 {|q|>1} and θ is an arithmetic function with suitable divisibility and growth conditions. As applications of our main theorem, we give linear independence results for various kinds of Lambert series.