Eigenvalues of Matrices whose Elements are Ramanujan Sums or Kloosterman Sums

TL;DR

This paper investigates the eigenvalues of matrices constructed from Ramanujan sums and Kloosterman sums, revealing their spectral properties and behaviors for fixed positive integers.

Contribution

It introduces a detailed analysis of eigenvalues for matrices with entries as Ramanujan or Kloosterman sums, including sums of these sums, which is a novel exploration in number theory.

Findings

Eigenvalues characterized for matrices with Ramanujan sums

Eigenvalues characterized for matrices with Kloosterman sums

Spectral properties depend on the structure of the sums

Abstract

Let $c_q(n)$ be the Ramanujan sums and let $S(m,n;q)$ be the Kloosterman sums. We study the eigenvalues of $q \times q$ matrices whose $(m,n)$ entry is $c_q (m-n)$ or $S(m,n;q)$ where $q$ is a fixed positive integer. We also study the eigenvalues of matrices whose entries are sums of Ramanujan sums or sums of Kloosterman sums.