For each prime
, let
denote the least quadratic
nonresidue modulo
.
Vinogradov conjectured that
for every fixed
.
This conjecture follows from the generalized Riemann hypothesis and is known to hold for almost
all primes
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
(when the conductor
is not cube-free) or
(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
or
at level
above
or
,
respectively. Some applications to the least primitive root are also given.
Keywords
quadratic nonresidue, Elliott–Halberstam conjecture,
character sums, Burgess bound
