p-adic Gauss integrals from the Poison summarizing formula

TL;DR

This paper introduces a novel method using Poisson summation, renormalization, and ergodic theories to derive classical formulas for Gauss sums, Gauss reciprocity, and p-adic Gauss integrals.

Contribution

It provides a new, simplified derivation of key formulas in number theory by combining Poisson summation with ergodic and renormalization techniques.

Findings

Derived Gauss sums and reciprocity law from Poisson summation.

Obtained a new simple derivation of p-adic Gauss integrals.

Unified approach connecting classical and p-adic number theory.

Abstract

In the present paper we show how to obtain the well-known formula for Gauss sums and the Gauss reciprocity low from the Poison summarizing formula by using some ideas of renormalization and ergodic theories. We also apply our method to obtain new simple derivation of the standard formula for p-adic Gauss integrals.