About a theorem of Paolo Codecà's and \(\Omega\)-estimates for arithmetical convolutions: Addendum
The condition of the author's Theorem 1 in [J. Number Theory 30, 71-85 (1988; Zbl 0649.10034)] that the parameter \(B\geq 0\) has to be an integer is now removed. This version of the theorem can be used to obtain two- sided \(\Omega\)-estimates for the class of convolutions \[ g(x):=\sum_{n\leq x^{3/4}}\alpha (n)n^ af(x/n), \] where a,x\(\in {\mathbb{R}}:\) \(a\leq -1\), \(x>0\), \(\alpha (n)=1\) or \(\alpha (n)=\mu (n)\) (Möbius function) and f() is a 1-periodic function satisfying a Kubert identity of order \(1\leq \ell \in {\mathbb{N}}:\) \[ f(my)=m^{\ell - 1}\sum^{m-1}_{n=0}f(y+n/m)\quad \forall y\in {\mathbb{R}},\quad m\in {\mathbb{N}}. \] As an application, the author proves new or improved \(\Omega_{\pm}\)-results for the divisor functions \(G_{a,k}(x)\) of Chowla and Walum, the P-function of Hardy and Littlewood and to an error term related to the Euler \(\phi\)-function. Moreover, the constants implied by the \(\Omega\)-symbol can be computed.
scientific article; zbMATH DE number 4210264