The authors study sufficient conditions for \(L^2\) and almost everywhere convergence of series of the form \[ \sum_k c_k f(kx), \] where \(f\) is a 1-periodic function, with \[ \int_0^1 f(x)\,dx=0, \quad \int_0^1 f^2(x)\,dx<\infty, \] and such that the Fourier coefficients \(a_n\), \(b_n\) of \(f\) have a decay of \(O(n^{-\alpha})\) (i.e., \(f\in C_\alpha\)). The case \(\alpha=1\) is covered in a previous work of the authors. Here the method is extended to handle the case \(\alpha \in (1/2, 1)\). Sharp (optimal) sufficient conditions are given in terms of convergence properties of the weighted \(c_k^2\) series. These convergence criteria have an arithmetic character. Furthermore, the authors construct examples for the divergence of more general series of the form \[ \sum_k c_k f(n_kx), \] using eigenvectors of GCD matrices belonging to the maximal eigenvalue, which are concentrated on indices \(k\) with many small prime factors.
The paper contains also an extensive survey of the relevant background material and of current lines of research.