Existence of all the asymptotic \(\lambda\)-th means for certain arithmetical convolutions
Let \(g: [1,\infty[\to \mathbb{R}\) be a function with the asymptotic expansion \[ g(x) = \sum_{n\leq z(x)} (a(n)/n)f(x_ n) + C + o(1)\qquad (x\to \infty;\quad C\in \mathbb{R}), \tag{*} \] where \(a(n)\) is a bounded sequence of real numbers satisfying \(\sum_{n\leq x} a(n) = Dx + o(x)\) with a constant \(D\in\mathbb{R}\), \(z(x)\nearrow \infty\) as \(x\to\infty\), but \(z(x)x^{-\varepsilon}\to 0\) for a ny \(\varepsilon>0\). Moreover \(f(\;)\) is a real-valued \(T\)-periodic function of bounded variation with \(\int^ T_0 f(t)dt =0\). Note that (*) is the `truncated' expansion for a wider class of functions \(g(\;)\) satisfying \[ g(x) = \sum_{n\leq x} (a(n)/n)f(x/n) + o(1). \] For \(0<\lambda \in \mathbb{R}\) and a function \(E:[1,\infty[\to\mathbb{R}\) the asymptotic \(\lambda\)-th mean of \(E\) is -- as long as the involved limit exists -- defined by \[ M(E,\lambda) := \lim_{x\to \infty} x^{-1} \int^ x_1 E^\lambda(t) dt. \] In general the author shows the existence of (a) \(M(g,k)\) for all \(k\in \mathbb{N}\), (b) \(M(| g|,\lambda)\) for all \(0<\lambda \in \mathbb{R}\) and (c) \(M(g-C,2\ell-1)=0\) for all \(\ell\in\mathbb{N}\), utilizing and refining some results of \textit{P. Codecà} [Rend. Semin. Mat. Univ. Padova 72, 373-387 (1984; Zbl 0552.10030)]. As an application of these results it is shown (Theorem 1) that the well-known (error-) functions \[ H(x) := \sum_{n\leq x} \phi(n)/n - (6/\pi^2)x,\qquad G(x) := \sum_{n\leq x} \sigma(n)/n - (\pi^2/6)x, \] \[ P(x) := \sum_{n\leq x} (1/n)\sin(x/n),\qquad Q(x) := \sum_{n\leq x} (1/n)\cos(x/n) \] have an asymptotic expansion of the requested type (*) -- with some \(z(\;)\) and \(C\) in each case -- hence one obtains easily the mean values (a)- (c) for these functions.
scientific article; zbMATH DE number 4079552