Upper Bound of the Least Quadratic Nonresidues

TL;DR

The paper addresses the problem of the least quadratic nonresidue $n(p)$ modulo a large prime $p$ and improves the unconditional upper bound from the Burgess-type estimate $n(p)\ll p^{\frac{1}{4\sqrt{e}}+\varepsilon}$ to $n(p)\ll (\log p)(\log\log p)^{1+\varepsilon}$, unconditionally. It introduces a new indicator function for quadratic nonresidues in finite fields and develops a finite-field Fourier-analytic framework, including a primitive-root representation and Gauss-sum inputs, to analyze the problem. The method combines a refined finite Fourier transform with summation kernels, bounds for incomplete exponential sums over consecutive and coprime indices, and a careful main-term/error-term decomposition to force a contradiction if no nonresidue lies below a logarithmic-threshold $x$. The results substantially tighten the nonresidue scale unconditionally, representing a significant step beyond exponential barriers and contributing toward the conjectured logarithmic bounds.

Abstract

Let $p\geq3$ be a large prime and let $n(p)\geq2$ denotes the least quadratic nonresidue modulo $p$. This note sharpens the standard upper bound of the least quadratic nonresidue from the unconditional upper bound $n(p)\ll p^{1/4\sqrt{e}+\varepsilon}$ to the conjectured upper bound $n(p)\ll (\log p)^{1+\varepsilon}$, where $\varepsilon>0$ is a small number, unconditionally. This improvement breaks the exponential upper bound barrier.

Theorems & Definitions (41)

• Conjecture 1.1
• Theorem 1.1
• Lemma 2.1
• Lemma 2.2
• proof : Proof
• Definition 3.1
• Definition 3.2
• Lemma 3.1
• proof : Proof
• Lemma 3.2