Multiplicative Functions with Sum Zero Over Beurling Generalized Prime Number Systems

Abstract

Completely multiplicative arithmetic functions with sum zero (in short CMO functions) were initially introduced by Kahane and Saïas (Expo Math 35(4):364–389, 2017). These functions were recently generalized to Beurling generalized prime number systems and denoted as $CMO_{\mathcal {P}}$ by Neamah (Res Number Theory 6:45, 2020). In this article, we generalize $CMO_{\mathcal {P}}$ to multiplicative functions and refer to them as $MO_{\mathcal {P}}$ functions. Thereafter, we examine some of these functions’ properties and provide examples of such functions. The aim was to investigate how small the partial sums of this class of functions can be. The results of this study indicate that for all $MO_{\mathcal {P}}$ functions with an abscissa of convergence 1, we have

$$\begin{aligned} \sum _{\begin{array}{c} n \le x \\ n\in \mathcal {N} \end{array}} f(n) = \Omega \Big (\frac{1}{\sqrt{x}} \Big ). \end{aligned}$$

We also discuss the consequences of this statement.