Algebraic Approaches to Periodic Arithmetical Maps |
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Authors: | Zhi-Wei Sun |
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Affiliation: | Department of Mathematics, Nanjing University, Nanjing, 210093, The People's Republic of Chinaf1 |
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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) = ∑ns|x − 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}|. |
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