On the \(B^{\lambda}\) almost periodic behaviour of certain arithmetical convolutions
The author is interested in the functions \[ P(x)=\sum_{n\leq x}(1/n)\cos (x/n), \quad Q(x)=\sum_{n\leq x}(1/n)\sin (x/n), \] \[ R_1(x)=-\sum_{n\leq x}(1/n)(\{x/n\}-1/2),\quad R_2(x)=-\sum_{n\leq x}(\mu (n)/n)(\{x/n\}- 1/2); \] the latter two are connected with the asymptotic formulas for \(\sum_{n\leq x}\sigma (n)/n\), resp. \(\sum_{n\leq x}\phi (n)/n\). All these functions are unbounded; however, as a corollary to his main theorems, the author shows that \(E(x)=E_1(x)+O(1),\) where \(E_1\) is \(B^{\lambda}\)-almost-periodic, and that \((1/x)\int^{x}_{1}| E(t)|^{\lambda} \, dt=O(1)\) for any \(\lambda >0\), where \(E(x)\) is one of the four functions \(P(x),\ldots,R_2(x)\). The author's main result is as follows: Define the convolution \[ g(x)=\sum_{n\leq y(x)}(a(n)/n)f(x/n), \] where \(a(n)\) is a bounded sequence of real numbers, \(y(x)\nearrow \infty\), as \(x\to \infty\), but \(y(x)x^{-\varepsilon}\to 0\) for any \(\varepsilon >0\), and where \(f\) is a real-valued 1-periodic function of bounded variation satisfying \(\int^{1}_{0}f(x) \,dx=0\). Then \(g\) is \(B^{\lambda}\)-almost-periodic for any \(\lambda >0\), and the mean value \[ M(g^k)=\lim_{x\to \infty}(1/x)\int^{x}_{1}g(t)^k \,dt \] exists for any integer \(k>0\).
scientific article; zbMATH DE number 3879029