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

TL;DR

This paper develops arithmetical methods to analyze the convergence of series of dilated functions, connecting it with properties of the Riemann Zeta function and improving existing convergence criteria.

Contribution

It introduces a new decomposition technique for squared sums of dilated functions and establishes refined convergence conditions involving arithmetical functions and the Riemann Zeta function.

Findings

Established bounds for series of dilated functions using arithmetical functions.

Improved convergence criteria for series of functions in BV space.

Connected convergence properties with the behavior of 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 $.