Convergence of series of dilated functions and spectral norms of GCD matrices

We establish a connection between the L2 norm of sums of dilated functions whose jth Fourier coefficients are O(j−α) for some α∈(1/2,1), and the spectral norms of certain greatest common divisor (GCD) matrices. Utilizing recent bounds for these spectral norms, we obtain sharp conditions for the convergence in L2 and for the almost everywhere convergence of series of dilated functions.