Abstract: This paper establishes new bridges between the class of complex functions, which contains zeta functions of arithmetic schemes and closed with respect to product and quotient, and the class of mean-periodic functions in several spaces of functions on the real line. In particular, the meromorphic continuation and functional equation of the Hasse zeta function of arithmetic scheme with its expected analytic shape is shown to imply the mean-periodicity of a certain explicitly defined function associated to the zeta function. Conversely, the mean-periodicity of this function implies the meromorphic continuation and functional equation of the zeta function. This opens a new road to the study of zeta functions via the theory of mean-periodic functions which is a part of modern harmonic analysis. The case of elliptic curves over number fields and their regular models is treated in more details, and many other examples are included as well.