A proof of the Landsberg–Schaar relation by finite methods

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.

Double-Janus linear sigma models and generalized reciprocity for Gauss sums

“…In Riemann and Mumford's notations, ϑ(z; τ ) ≡ ϑ00(z; τ ) is denoted as θ3(z; q), where q ≡ e 2πiτ 2. The Landsberg-Schaar identity was also recently proved using a non-analytical method[34]. For the history of development of various proofs other than Gauss's after 1801, see[35].…”