Quadratic nonresidues below the Burgess bound

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