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

Abstract

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 $d(m)=\sum_{d|m} 1$, and let $f_n(x) = f(nx)$. We show by using a new decomposition of squared sums that for any $K\subset \N$ 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)$. 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 $0< \e\le 2s-1$, $\zeta(2s)^{-1} \|\sum_{k\in K} c_k f^s_k \|_2^2 \le \frac{1+\e}{\e } (\sum_{k \in K} |c_k|^2 \s_{ 1+\e-2s}(k) )$, where $\s_h(n)=\sum_{d|n}d^h$. From this we deduce that if $f\in {\rm BV}(\T)$, $\langle f,1\rangle=0$ and $$\sum_{k} c_k^2\frac{(\log\log k)^4}{(\log\log \log k)^2} <\infty,$$ then the series $ \sum_{k } c_kf_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 $ converges for some $\g>4$. We further show that the same conclusion holds under the arithmetical condition $$\sum_{ k } c_k^2 (\log\log k)^{2 + b} \s_{ -1+\frac{1}{(\log\log k)^{ b/3}} }(k) <\infty,$$ for some $b>0$, or if $ \sum_{ k} c_k^2 d(k^2) (\log\log k)^2 <\infty$. We also derive from a recent result of Hilberdink an $\O$-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}^\infty c_k f_k $.