An elementary proof of the Selberg identity for Kloosterman sums

TL;DR

This paper presents a straightforward proof of the Selberg identity for Kloosterman sums, relying solely on the orthogonality of additive characters, simplifying previous approaches.

Contribution

It provides an elementary proof of the Selberg identity for Kloosterman sums using basic properties of additive characters.

Findings

Simplified proof of the Selberg identity

Relies only on orthogonality of additive characters

Accessible approach to a classical result

Abstract

We give an elementary proof of the Selberg identity for Kloosterman sums, which only requires the orthogonality of additive characters.