Limit-periodic arithmetical functions and the ring of finite integral adeles (Q392976)

An interesting connection between limit-periodic arithmetical functions and random variables on the probability space of finite integral adèles was established in work of \textit{E. V. Novoselov} [Izv. Akad. Nauk SSSR, Ser. Mat. 28, 307--364 (1964; Zbl 0213.33502)]. The paper under review brings further contributions to this line of research. Let \(\mathfrak D\) denote the space of periodic arithmetical functions, linearly spanned by the exponential function \(e_\alpha (n)=\exp (2\pi i\alpha n)\), \(\alpha \in {\mathbb Q}/{\mathbb Z}\). For each \(q\in [1,\infty)\), consider the separated completion \(D^q\) of \(\mathfrak D\) with respect to the Besicovitch seminorm \(\| f\|_q :=\limsup_N \big( \frac{1}{N} \sum_{n=1}^N | f(n) |^q \big)^{1/q}\). The natural diagonal embedding of \(\mathbb N\) into the ring \(\widehat{\mathbb Z}\) of finite integral adèles allows the functions \(e_\alpha\) to extend uniquely to characters \(\hat{e}_\alpha\) of the compact group \((\widehat{\mathbb Z},+)\). The family \((\hat{e}_\alpha)_{\alpha\in {\mathbb Q}/{\mathbb Z}}\) forms an orthonormal basis of the Hilbert space \(L^2(\widehat{\mathbb Z}, \mathrm{ Bor}_{\widehat{\mathbb Z}},\lambda)\). Furthermore, the additive Haar measure \(\lambda\) on \(\widehat{\mathbb Z}\) coincides with the \(w^*\)-limit of the sequence of probability measures \(\big( \frac{1}{N} \sum_{n=1}^N \delta_n \big)_N\). One result in this paper is that the finite Fourier expansion \[ S_n (f):= \sum_{r\mid n} \sum\limits_{1\leq a\leq r, (a,r)=1} \langle f, e_{a/r}\rangle e_{a/r} \] of every \(f\in D^q\), \(q\in [1,\infty)\), converges to \(f\) in \(D^q\) as \(n\rightarrow 0\) in \(\widehat{\mathbb Z}\). The proof employs standard ideas from harmonic analysis and interpolation. An interesting application to Ramanujan-Fourier expansions shows that a similar result holds when \(f\) is a \(q\)-almost-even arithmetical function and \(S_n (f)\) is replaced by \(\sum_{r\mid n} \varphi(r)^{-1} \langle f,c_r\rangle c_r\), where \(c_r\) denotes the Ramanujan sum \(\sum_{1\leq a\leq r, (a,r)=1} e_{a/r}\). This extends Theorem VI.5.1 in the monograph on Arithmetical Functions of \textit{W. Schwarz} and \textit{J. Spilker} [Arithmetical functions. An introduction to elementary and analytic properties of arithmetic functions and to some of their almost-periodic properties. London Mathematical Society Lecture Note Series. 184. Cambridge: Cambridge University Press. (1994; Zbl 0807.11001)], a result originally proved by A. J. Hildebrand [Acta Arith. 44, 110-140 (1984;Zbl 0507.10001)]. Another result concerns multiplicative \(q\)-limit-periodic functions \(f(n)=\prod_p f_p(n)\). Each factor \(f_p\) extends to a random variable \(\hat{f}_p\) on \(\widehat{\mathbb Z}\) which is constant on \(p^k \widehat{\mathbb Z} \setminus p^{k+1} \widehat{\mathbb Z}\), \(k=0,1,\ldots\) It is shown that if the mean value \(M[f]:=\lim_N \frac{1}{N} \sum_{n=1}^N f(n)\) is \(\neq 0\), then the product \(\hat{f}(x):=\prod_p \hat{f}_p (x)\) of the independent random variables \(\hat{f}_p\) converges \(\lambda\) a.e. on \(\widehat{\mathbb Z}\). A key ingredient in the proof is Doob's martingale convergence theorem. The similar result for additive functions was established by Novoselov [loc. cit].