Arithmetic Poisson summation formulas
Abstract
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.
Related
- A Classification of Fourier Summation Formulas and Crystalline Measures
- A unification of Euler-Maclaurin and Euler-Boole summation formulas
- ABRR Summation Formulas for Relative Extremal Projectors
- Classification of Fourier summation formulas on a horizontal strip
- A Degenerate Clausenian 3F2(1) Summation and Its Arithmetic Specialization
BibTeX
@misc{panzone2026arithmetic,
title = {{Arithmetic Poisson summation formulas}},
author = {Pablo Panzone and Lucas Oliveira},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36606},
eprint = {2610.36606},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36606},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Arithmetic Poisson summation formulas
AU - Pablo Panzone
AU - Lucas Oliveira
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.36606
UR - https://arxiv.org/abs/2610.36606
AB - 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 x^2+y^2+1 and x^2+y^2+z^2+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 x_1^2+⋯+x_n^2+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.
ER -