Arithmetical functions and irrationality of Lambert series

We use a method of Erdos in order to prove the linear independence over Q of the numbers for every q member of Z, with |q| greater than or equal to 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.