Vol. 18, No. 1, 1966

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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
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

F (mn )F((m,n )) = F(m)F (n )f((m,n ))

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
Authors
James E. Shockley