Abstract :
A residue class a + n with weight λ is denoted by λ, a, n . For a finite system = { λs, as, ns }ks = 1 of such triples, the periodic map w (x) = ∑nsx − as λs is called the covering map of . Some interesting identities for those with a fixed covering map have been known; in this paper we mainly determine all those functions f : Ω → such that ∑ks = 1 λsf(as + ns ) depends only on w where Ω denotes the family of all residue classes. We also study algebraic structures related to such maps f, and periods of arithmetical functions ψ(x) = ∑ks = 1 λse2πiasx/ns and ω(x) = {1 ≤ s ≤ k : (x + as, ns) = 1}.
