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A generalization of Ramanujan's sum over finite rings

Priya, Sanjay Kumar Singh

TL;DR

The paper addresses the problem of determining eigenvalues of the unitary Cayley graph \(\operatorname{Cay}(R, R^{\times})\) for arbitrary finite rings, generalizing Ramanujan sums beyond the classical \(\mathbb{Z}_n\) case. It expresses eigenvalues as additive-character sums \(C_{\alpha}(x) = \sum_{s\in (x)_{\ell}} \psi_{\alpha}(s)\) and derives an explicit formula using Möbius inversion on left ideals, parameterized by the largest left ideal \(K\) contained in \((x)_{\ell} \cap \langle \alpha \rangle^{\perp}\). A central result shows that, under a minimality condition, \(C_{\alpha}(x) = \mu_R(K,(x)_{\ell}) \dfrac{|[x]_{\ell}|}{\varphi_R(K,(x)_{\ell})}\), with a commutative special case mirroring the classical Ramanujan sum. The work extends to right and two-sided ideals and lays groundwork for spectral analysis of Cayley graphs over noncommutative rings, offering a unified framework that subsumes Ramanujan sums as a special instance. This provides a precise, algebraically grounded method to compute eigenvalues of unitary Cayley graphs over finite rings, enhancing both number-theoretic and graph-theoretic applications.

Abstract

Let $R$ be a finite ring with unity. In general, the eigenvalues of the unitary Cayley graph $\text{Cay}(R, R^{\times})$ are not known when $R$ is a non-commutative. In this paper, we present an explicit formula for the eigenvalues of $\text{Cay}(R, R^{\times})$ for any finite ring $R$. However, our focus is on a more general case of the unitary Cayley graph. It is well known that the classical Ramanujan's sum represents the eigenvalues of $\text{Cay}(\mathbb{Z}_n, \mathbb{Z}_n^{\times})$. Consequently, the eigenvalues of $\text{Cay}(R, R^{\times})$ can be view as a generalization of classical Ramanujan's sum in the context of finite rings. Interestingly, the formula we derive for the eigenvalues of $\text{Cay}(R, R^{\times})$ extends the known formula of classical Ramanujan's sum to the context of finite rings.

A generalization of Ramanujan's sum over finite rings

TL;DR

The paper addresses the problem of determining eigenvalues of the unitary Cayley graph \(\operatorname{Cay}(R, R^{\times})\) for arbitrary finite rings, generalizing Ramanujan sums beyond the classical case. It expresses eigenvalues as additive-character sums \(C_{\alpha}(x) = \sum_{s\in (x)_{\ell}} \psi_{\alpha}(s)\) and derives an explicit formula using Möbius inversion on left ideals, parameterized by the largest left ideal contained in \((x)_{\ell} \cap \langle \alpha \rangle^{\perp}\). A central result shows that, under a minimality condition, \(C_{\alpha}(x) = \mu_R(K,(x)_{\ell}) \dfrac{|[x]_{\ell}|}{\varphi_R(K,(x)_{\ell})}\), with a commutative special case mirroring the classical Ramanujan sum. The work extends to right and two-sided ideals and lays groundwork for spectral analysis of Cayley graphs over noncommutative rings, offering a unified framework that subsumes Ramanujan sums as a special instance. This provides a precise, algebraically grounded method to compute eigenvalues of unitary Cayley graphs over finite rings, enhancing both number-theoretic and graph-theoretic applications.

Abstract

Let be a finite ring with unity. In general, the eigenvalues of the unitary Cayley graph are not known when is a non-commutative. In this paper, we present an explicit formula for the eigenvalues of for any finite ring . However, our focus is on a more general case of the unitary Cayley graph. It is well known that the classical Ramanujan's sum represents the eigenvalues of . Consequently, the eigenvalues of can be view as a generalization of classical Ramanujan's sum in the context of finite rings. Interestingly, the formula we derive for the eigenvalues of extends the known formula of classical Ramanujan's sum to the context of finite rings.
Paper Structure (5 sections, 15 theorems, 53 equations)

This paper contains 5 sections, 15 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. M$\ddot{\text{o}}$bius Inversion Formula
  4. Main Theorem
  5. Conclusion

Key Result

Theorem 1.1

Let $Z$ be an abelian group and $S$ be a close under inverse subset of $Z$. The eigenvalue of Cayley graph $\operatorname{Cay}(Z, S)$ is $\{ \lambda_\alpha \colon \alpha \in Z \}$, where

Theorems & Definitions (27)

  • Theorem 1.1: babai1979spectra
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • ...and 17 more