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 ${\sum }_k{c}_{k}f(kx)$. Let $f(x)= \sum_{m=1}^\infty a_m \sin {2\pi m x}$where$\sum_{m=1}^\infty a_{m }^2d(m) <\infty$and$d(m)=\sum_{d|m} 1$,andlet$f_n(x) = f(nx)$. We show by using a new decomposition of squared sums that for any $K\subset \mathbb{N}$ finite, $ |\sum_{k\in K} c_k f_k |2^2 \le ( \sum{m=1}^\infty a_{m }^2 d(m) ) \sum_{k\in K } c_{k}^2d(k^2)$\mathrm{.}If$f^s (x)= \sum_{j=1}^\infty \frac{\sin 2\pi jx}{j^s}$,$s>1/2$, by only using elementary Dirichlet convolution calculus, we show that for $0<\text{e}\le 2s-1$, $\zeta(2s)^{-1} |\sum_{k\in K} c_k f^s_k |2^2 \le \frac{1+\e}{\e } (\sum{k \in K} |c_k|^2 \s_{ 1+\e-2s}(k) )$,where$\s_h(n)=\sum_{d|n}d^h$. From this we deduce that if $f\in \mathrm{B}\mathrm{V}(\text{T})$, $⟨f,1⟩=0$ and $$\sum _k{c}_k^2\frac{(\log \log k{)}^4}{(\log \log \log k{)}^2}<\mathrm{\infty },$$ then the series ${\sum }_k{c}_k{f}_k$ converges almost everywhere. This slightly improves a recent result, depending on a fine analysis on the polydisc (\cite{ABS}, th.3) ($n_k=k$), where it was assumed that $ \sum_{k} c_k^2 , (\log\log k)^\g $convergesforsome$\g>4$. We further show that the same conclusion holds under the arithmetical condition $$\sum _k{c}_k^2(\log \log k{)}^{2+b}{\text{s}}_{-1+\frac{1}{(\log \log k{)}^{b\mathrm{/}3}}}(k)<\mathrm{\infty },$$ for some $b>0$, or if $ \sum_{ k} c_k^2 d(k^2) (\log\log k)^2 <\infty$\mathrm{.}WealsoderivefromarecentresultofHilberdinkan$\O$-result for the Riemann Zeta function involving factor closed sets. We finally prove an important complementary result to Wintner’s famous characterization of mean convergence of series ${\sum }_{k=0}^{\mathrm{\infty }}{c}_k{f}_k$.