Arithmetic Poisson summation formulas

We formulate Poisson summation with periodic weights in forms adapted to arithmetic sums (arithmetic Poisson summation formulas, APSF), in scalar, polynomial-congruence and vector-divisibility versions, extending Berndt's arithmetic Poisson formula. We use APSF as a common framework to rederive Ramanujan's formula for $ζ(2m+1)$ and its character analogues due to Katayama and Berndt-Straub, the functional equations of $ζ$, of Dirichlet's beta function and of Estermann's function, and the exponents of Tolev and Chen-Wang for power-free values of x²+y²+1 and x²+y²+z²+k. It also gives reciprocity laws with Kelvin-function kernels for Lambert series twisted by a Dirichlet character over imaginary quadratic fields of arbitrary class number; asymptotic formulas with error terms for r-free values of x₁²+⋯+xₙ²+k, with Gaussian-weighted versions and the meromorphic continuation of the associated Dirichlet series; an argument of Terras, relating two-dimensional Bessel series for all odd zeta values and even Dirichlet beta values; and a Gaussian representation of a Mersenne-Fermat prime detector.