The Elliott–Halberstam conjecture implies the Vinogradov least quadratic nonresidue conjecture

Abstract

For each prime $p$, let $n\left(p\right)$ denote the least quadratic nonresidue modulo $p$. Vinogradov conjectured that $n\left(p\right)=O\left(p^{\epsilon }\right)$ for every fixed $\epsilon >0$. This conjecture follows from the generalized Riemann hypothesis and is known to hold for almost all primes $p$ but remains open in general. In this paper, we show that Vinogradov’s conjecture also follows from the Elliott–Halberstam conjecture on the distribution of primes in arithmetic progressions, thus providing a potential “nonmultiplicative” route to the Vinogradov conjecture. We also give a variant of this argument that obtains bounds on short centered character sums from “Type II” estimates of the type introduced recently by Zhang and improved upon by the Polymath project or from bounds on the level of distribution on variants of the higher-order divisor function. In particular, an improvement over the Burgess bound would be obtained if one had Type II estimates with level of distribution above $\frac{2}{3}$ (when the conductor is not cube-free) or $\frac{3}{4}$ (if the conductor is cube-free); morally, one would also obtain such a gain if one had distributional estimates on the third or fourth divisor functions ${\tau }_3$ or ${\tau }_4$ at level above $\frac{2}{3}$ or $\frac{3}{4}$, respectively. Some applications to the least primitive root are also given.