  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.
These formulas simultaneously generalize the classical Poisson formula and
Newton formulas for Newton sums. Classical Poisson formulas in Fourier
analysis, classical summation formulas as Euler-McLaurin or Abel-Plana
formulas, explicit formulas in number theory and Selberg trace formulas in
Riemannian geometry appear as special cases of our general Poisson-Newton
formula. We also associate to finite order meromorphic functions general
Poisson-Newton formulas that yield many classical integral formulas.
