Arithmetical aspects of certain functional equations

The classical system of functional equations ${\displaystyle 1/n{\sum }_{\nu =0}^{n-1}F((x+\nu )/n)=n^{-s}F\left(x\right)}$ (n ∈ ℕ) with s ∈ ℂ, investigated for instance by Artin (1931), Yoder (1975), Kubert (1979), and Milnor (1983), is extended to ${\displaystyle 1/n{\sum }_{\nu =0}^{n-1}F((x+\nu )/n)={\sum }_{d=1}^{\infty }{\lambda }_n\left(d\right)F\left(dx\right)}$ (n ∈ ℕ) with complex valued sequences ${\displaystyle {\lambda }_n}$. 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.