The convergence of certain Diophantine series

For x irrational, we study the convergence of series of the form ∑ n − s f ( n x ) where f is a real-valued, 1-periodic function which is continuous, except for singularities at the integers with a potential growth. We show that it is possible to fully characterize the convergence set and to approximate the series in terms of the continued fraction of x. This improves and generalizes recent results by Rivoal who studied the examples f ( t ) = cot ⁡ ( π t ) and f ( t ) = sin − 2 ⁡ ( π t ) .