# 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.

## Key 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.