Abstract/Summary

CMOfunctions are completely multiplicative functionsffor which∑∞n=1f(n) = 0.Such functions were first defined and studied by Kahane and Sa ̈ıas [31]. We extendthese to multiplicative functions with the aim to investigate the theory of the extendedfunctions and we shall call themMOfunctions. We give some properties and findexamples ofMOfunctions, as well as pointing out the connection between thesefunctions and the Riemann hypothesis at the end of Chapter 2.Following this, we broaden our scope by generalising bothCMOandMOfunctionsto Beurling prime systems with the aim to answer some of the questions which wereraised by Kahane and Sa ̈ıas. We shall call these sets of functionsCMOPandMOPfunctions. Such generalisations allow us to look for new examples to extend ourknowledge. In particular, we explore how quickly the partial sum of these classesof functions tends to zero with different Beurling generalised prime systems. Thefindings of this study suggest that for allCMOPandMOPfunctionsfoverNwithabscissa 1, we have ∑n≤xn∈Nf(n) = Ω(1√x).