A proof of the Landsberg-Schaar relation by finite methods

TL;DR

This paper provides a direct, analysis-free proof of the Landsberg-Schaar relation, a key identity involving quadratic Gauss sums, using finite methods instead of traditional limiting approaches.

Contribution

It introduces a novel finite-method proof of the Landsberg-Schaar relation, bypassing the usual analytical techniques involving theta functions.

Findings

Establishes a new finite-method proof of the Landsberg-Schaar relation.

Simplifies understanding of quadratic Gauss sum identities.

Avoids complex analysis in proving classical identities.

Abstract

The Landsberg-Schaar relation is a classical identity between quadratic Gauss sums, normally used as a stepping stone to prove quadratic reciprocity. The Landsberg-Schaar relation itself is usually proved by carefully taking a limit in the functional equation for Jacobi's theta function. In this article we present a direct proof, avoiding any analysis.