2021/08/26 by Neamah, Ammar Ali
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2108.11937
CMO functions are completely multiplicative functions f for which ∑n=1^∞ f(n) =0. These functions were first introduced and studied by Kahane and Saïas [5]. The main purpose of this paper is to generalise such functions to multiplicative functions and we shall call them MO functions. More precisely, we define MO functions to be multiplicative functions for which ∑n=1^∞ f(n) = 0 and ∑k=0^∞ f(pk) ≠ 0 for all p ∈ ℙ. We give some properties and find examples of MO functions, as well as pointing out the connection between these functions and the Riemann hypothesis at the end of the paper.