Completely multiplicative functions with sum zero over generalised prime systems

Abstract

CMO functions multiplicative functions f for which sum _{n=1}^infty f(n) =0. Such functions were first defined and studied by Kahane and Saïas [14]. We generalised these to Beurling prime systems with the aim to investigate the theory of the extended functions and we shall call them CMO_{mathcal {P}} functions. We give some properties and find examples of CMO_{mathcal {P}} functions. In particular, we explore how quickly the partial sum of these classes of functions tends to zero with different generalised prime systems. The findings of this paper may suggest that for all CMO_{mathcal {P}} functions f over mathcal{{N}} with abscissa 1, we have ∑n≤xn∈Nf(n)=Ω(1x).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\sum _{\\tiny \\begin{array}{c} n \\le x \\\\ n\\in \\mathcal {N} \\end{array}} f(n) = \\Omega \\Big (\\frac{1}{\\sqrt{x}} \\Big ). \\end{aligned}$$\\end{document}