Small prime $k$th power residues for $k=2,3,4$: A reciprocity laws approach

TL;DR

This paper improves bounds on the distribution of small prime residues for quadratic, cubic, and biquadratic cases using reciprocity laws, showing many such residues are smaller than previously established bounds.

Contribution

It introduces a reciprocity law-based method to establish lower bounds on the number of small prime power residues, avoiding character sum estimates.

Findings

For primes p ≡ 1 mod 3, many prime cubic residues are less than p^{1/2+ε}.

There are more than p^{1/9} prime quadratic residues less than p for large p.

The method applies similarly to biquadratic residues, extending classical results.

Abstract

Nagell proved that for each prime $p\equiv 1\pmod{3}$, $p > 7$, there is a prime $q<2p^{1/2}$ that is a cubic residue modulo $p$. Here we show that for each fixed $\epsilon > 0$, and each prime $p\equiv 1\pmod{3}$ with $p > p_0(\epsilon)$, the number of prime cubic residues $q < p^{1/2+\epsilon}$ exceeds $p^{\epsilon/30}$. Our argument, like Nagell's, is rooted in the law of cubic reciprocity; somewhat surprisingly, character sum estimates play no role. We use the same method to establish related results about prime quadratic and biquadratic residues. For example, for all large primes $p$, there are more than $p^{1/9}$ prime quadratic residues $q<p$.