An arithmetical approach to the convergence problem of series of dilated functions and its connection with the Riemann Zeta function

Given a periodic function f, we study the convergence almost everywhere and in norm of the series ∑kckf(kx). Let f(x)=∑m=1∞amsin⁡2πmx where ∑m=1∞am2d(m)<∞ and 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, ‖∑k∈Kckfk‖22≤(∑m=1∞am2d(m))∑k∈Kck2d(k2). If f(s)(x)=∑j=1∞sin⁡2πjxjs, s>1/2, by only using elementary Dirichlet convolution calculus, we show that for 0<ε≤2s−1, ζ(2s)−1‖∑k∈Kckfk(s)‖22≤1+εε(∑k∈K|ck|2σ1+ε−2s(k)), where σh(n)=∑d|ndh. From this we deduce that if f∈BV(T), 〈f,1〉=0 and∑kck2(log⁡log⁡k)4(log⁡log⁡log⁡k)2<∞, then the series ∑kckfk converges almost everywhere. This slightly improves a recent result, depending on a fine analysis on the polydisc [1, th. 3] (nk=k), where it was assumed that ∑kck2(log⁡log⁡k)γ converges for some γ>4. We further show that the same conclusion holds under the arithmetical condition∑kck2(log⁡log⁡k)2+bσ−1+1(log⁡log⁡k)b/3(k)<∞, for some b>0, or if ∑kck2d(k2)(log⁡log⁡k)2<∞. 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 havemax1≤t≤T⁡|ζ(σ+it)|≥C(σ)(1σ−2ε(ν)∑n|νσ−s+ε(n)2n2ε)1/2. We finally prove an important complementary result to Wintner's famous characterization of mean convergence of series ∑k=0∞ckfk.