Quadratic nonresidues below the Burgess bound

Abstract

For any odd prime number $p$, let $(\cdot |p)$ be the Legendre symbol, and let $n_1(p)<n_2(p)<\cdots$ be the sequence of positive nonresidues modulo $p$, i.e., $(n_k|p)=-1$ for each $k$. In 1957, Burgess showed that the upper bound $n_1(p){\ll }_{\epsilon }{p}^{(4\sqrt{e}{)}^{-1}+\epsilon }$ holds for any fixed $\epsilon >0$. In this paper, we prove that the stronger bound $$n_k(p)\ll p^{(4\sqrt{e}{)}^{-1}}\exp \left(\sqrt{e^{-1}\log p\log \log p}\right)$$ holds for all odd primes $p$, where the implied constant is absolute, provided that $$k\le p^{(8\sqrt{e}{)}^{-1}}\exp (\frac{1}{2}\sqrt{e^{-1}\log p\log \log p}-\frac{1}{2}\log \log p).$$ For fixed $\epsilon \in (0,\frac{\pi -2}{9\pi -2}]$ we also show that there is a number $c=c(\epsilon )>0$ such that for all odd primes $p$ and either choice of $\theta \in \{\pm 1\}$, there are ${\gg }_{\epsilon }y/(\log y{)}^{\epsilon }$ natural numbers $n\le y$ with $(n|p)=\theta$ provided that $$y\ge p^{(4\sqrt{e}{)}^{-1}}\exp (c(\log p{)}^{1-\epsilon }).$$

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