The author investigates the arithmetic origin and structure of functional equations of the type \[ {1\over n} \sum^{n-1}_{k=0} G(e^{2\pi ik/n}z) =\sum^\infty_{d=1} \lambda_n (d)G(z^{nd}) \] with natural \(n\) and complex \(z\), and the closely related equations \[ {1\over n} \sum^{n-1}_{k=0} F\left({x+k \over n} \right) =\sum^\infty_{d=1} \lambda_n (d)F (dx) \] with real \(x\). Some fundamental results concerning their holomorphic, their periodic integrable and their aperiodic continuous solutions, respectively are established. The main tools are of number-theoretic and functional-analytic nature.