  This paper establishes new bridges between number theory and modern harmonic
analysis, namely 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 zeta function of arithmetic scheme with its expected
analytic shape is shown to correspond to mean-periodicity of a certain
explicitly defined function associated to the zeta function. This
correspondence can be viewed as an extension of the Hecke--Weil correspondence.
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.
