Algebraic Approaches to Periodic Arithmetical Maps

AbstractA residue class a+nZ with weight λ is denoted by 〈λ,a,n〉. For a finite system A={〈λs,as,ns〉}ks=1 of such triples, the periodic map wA(x)=∑ns|x−asλs is called the covering map of A. Some interesting identities for those A with a fixed covering map have been known; in this paper we mainly determine all those functions f:Ω→C such that ∑ks=1λsf(as+nsZ) depends only on wA 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=1λse2πiasx/ns and ω(x)=|{1≤s≤k:(x+as,ns)=1}|