Algebraic approaches to periodic arithmetical maps

A residue class a + nZ with weight h is denoted by (lambda, a, n). For a fi nite system A = {[lambda (s), a(s), n(s)]}(s=1)(k) of such triples, the per iodic map omega (A)(x) = Sigma (ns\x-as) lambda (s) is called the covering map of A. Some interesting identities for those sl with a fixed covering ma p have been known; in this paper we mainly determine all those functions f : Omega --> C such that Sigma (k)(s=1) lambda (s)f(a(s) + n(s)Z) depends on ly on omega (A) where Omega denotes the family of all residue classes. We a lso study algebraic structures related to such maps f, and periods of arith metical functions psi (x) = Sigma (k)(s=1) lambda (s)e(s)(s)(2 pi ia)(x/n) and omega (x) = \{1 less than or equal to s less than or equal to k : (x a(s), n(s)) = 1}\. (C) 2001 Academic Press.