Arithmetical functions and irrationality of Lambert series

We use a method of Erdös in order to prove the linear independence over ℚ of the numbers for every $q∈Z,$ with $|q|≥2.$ The main idea consists in considering the two above series as Lambert series. This allows to expand them as power series of 1/q. The Taylor coefficients of these expansions are arithmetical functions, whose properties allow to apply an elementary irrationality criterion, which yields the result.