Quadratic nonresidues below the Burgess bound

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 $n_1(p)\ll_εp^{(4\sqrt{e})^{-1}+ε}$ holds for any fixed $ε>0$. In this paper, we prove that the stronger bound $n_k(p)\ll p^{(4\sqrt{e})^{-1}}\exp\big(\sqrt{e^{-1}\log p\log\log p}\,\big)$ holds for all odd primes $p$, where the implied constant is absolute, provided that $k\le p^{(8\sqrt{e})^{-1}} \exp\big(\tfrac12\sqrt{e^{-1}\log p\log\log p}-\tfrac12\log\log p\big).$ For fixed $ε\in(0,\frac{π-2}{9π-2}]$ we also show that there is a number $c=c(ε)>0$ such that for all odd primes $p$ and either choice of $θ\in\{\pm 1\}$, there are $\gg_εy/(\log y)^ε$ natural numbers $n\le y$ with $(n|p)=θ$ provided that $y\ge p^{(4\sqrt{e})^{-1}}\exp\big(c(\log p)^{1-ε}\big).$

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