Some results on arithmetical functions (Q2769707)

The exponential functions \(n\mapsto \exp(2\pi i\frac{a}{r}n)\), \(a,r\in \mathbb{N}\), \(a\leq r\), \((a,r)=1\) generate the vector space \({\mathcal D}\) of all periodic functions. Let NEWLINE\[NEWLINE\|f\|_q:= (\limsup_{x\to\infty} \frac{1}{x} \sum_{n\leq x}|f(n)|^q)^{1/q},\;q\in \mathbb{R},\;q\geq 1.NEWLINE\]NEWLINE The closure of \({\mathcal D}\) with respect to the semi-norm \(\|\cdot\|_q\) is the space \({\mathcal D}^q\) of \(q\)-limit-periodic functions. The author treats two kinds of problems. NEWLINENEWLINENEWLINE1. Which conditions on \(f\in {\mathcal D}^q\) imply \(\frac{1}{f}\in {\mathcal D}^q\) (theorems on inverses)? 2. Which conditions on \(f\in{\mathcal D}^q\) and \(G:\mathbb{N}\to \mathbb{N}\) imply \(f\circ G\in{\mathcal D}^q\) (theorems of composed functions)?NEWLINENEWLINENEWLINEHis theorems generalize and unify known results.NEWLINENEWLINEFor the entire collection see [Zbl 0968.00043].