Quadratic nonresidues below the Burgess bound

Abstract

For any odd prime number $p$, let $(\cdot|p)$ be the Legendre symbol, and let $n_1(p) 0$. 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 $$ y\ge p^{(4\sqrt{e})^{-1}}\exp\big(c(\log p)^{1-ε}\big). $$