Given a periodic function f , we study the convergence almost everywhere and in norm of the series ∑kckf(kx). Let where and 47a820949d1abcfbe" title="Click to view the MathML source">d(m)=∑d|m1, and let fn(x)=f(nx). We show by using a new decomposition of squared sums that for any K⊂N finite, . If , s>1/2, by only using elementary Dirichlet convolution calculus, we show that for 0<ε≤2s−1, , where σh(n)=∑d|ndh. From this we deduce that if 47a949a0c8e75b74174" title="Click to view the MathML source">f∈BV(T), 〈f,1〉=0 and
then the series ∑kckfk converges almost everywhere. This slightly improves a recent result, depending on a fine analysis on the polydisc [1, th. 3] (a401b" title="Click to view the MathML source">nk=k), where it was assumed that converges for some γ>4. We further show that the same conclusion holds under the arithmetical condition
for some b>0, or if . We also derive from a recent result of Hilberdink an Ω-result for the Riemann Zeta function involving factor closed sets. As an application we find that simple conditions on T and ν ensuring that for any σ>1/2, 0≤ε<σ, we have
We finally prove an important complementary result to Wintner's famous characterization of mean convergence of series .