An attempt to interpret the Weil explicit formula from Beurling's spectral theory

A. Beurling introduced the celebrated problem of spectral synthesis. Roughly speaking, it is a problem whether functions belonging to a certain Banach space have a possibility to be approximated by trigonometric polynomials in the appropriate topology. For this problem Beurling introduced the concept of spectral sets whose elements are regarded as exponents of trigonometric polynomials. In the Weil explicit formula we can see a certain phenomenon which may be related to Beurling's spectral sets. The purpose of this paper is to study the phenomenon.