  We prove that a Poisson-Newton formula, in a broad sense, is associated to
each Dirichlet series with a meromorphic extension to the whole complex plane
of finite order. These formulas simultaneously generalize the classical Poisson
formula and Newton formulas for Newton sums. Classical Poisson formulas in
Fourier analysis, explicit formulas in number theory and Selberg trace formulas
in Riemannian geometry appear as special cases of our general Poisson-Newton
formula.
