A note on Benli and Pollack's paper on small prime power residues

TL;DR

This paper revisits and simplifies a proof regarding the count of small prime power residues, strengthening the bounds established by Benli and Pollack using reciprocity laws.

Contribution

It provides a simplified proof and slightly improved bounds on the number of small prime k-th power residues compared to prior work.

Findings

Simplified proof of bounds on small prime power residues

Stronger bounds established for the number of such residues

Utilizes reciprocity laws in the proof approach

Abstract

In this note, we revisit a result of Benli's and Pollack's on the number of small prime $k^{th}$ power residues. The proof is based on their idea of using reciprocity laws, but the argument is simplified and we prove a slightly stronger bound.