On the functional equation F(mn)F((m, n)) = F(m)F(n)f((m,n))

James E. Shockley

Vol. 18 (1966), No. 1, 185–189

DOI: 10.2140/pjm.1966.18.185

Abstract

Let f be a multiplicative arithmetic function, f(1) = 1. Necessary and sufficient conditions on f will be found so that the functional equation

will have a solution F with F(1)≠0 and all solution F will be determined. It will be shown that two different types of solutions may exist and that one of these requires that f have a property similar to complete multiplicativity.

Mathematical Subject Classification

Primary: 10.43

Milestones

Received: 23 October 1964

Revised: 24 February 1965

Published: 1 July 1966