Arithmetical aspects of certain functional equations
Volume 82 / 1997
Acta Arithmetica 82 (1997), 257-277
DOI: 10.4064/aa-82-3-257-277
Abstract
The classical system of functional equations $ 1/n ∑_{ν=0}^{n-1} F((x+ν)/n) = n^{-s} F(x)$ (n ∈ ℕ) with s ∈ ℂ, investigated for instance by Artin (1931), Yoder (1975), Kubert (1979), and Milnor (1983), is extended to $ 1/n ∑_{ν=0}^{n-1} F((x+ν)/n) = ∑_{d=1}^∞ λ_n(d)F(dx)$ (n ∈ ℕ) with complex valued sequences $λ_n$. This leads to new results on the periodic integrable and the aperiodic continuous solutions F:ℝ₊ → ℂ interrelating the theory of functional equations and the theory of arithmetic functions.
