Vol. 9, No. 4, 2015

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
The Elliott–Halberstam conjecture implies the Vinogradov least quadratic nonresidue conjecture

Terence Tao

Vol. 9 (2015), No. 4, 1005–1034

For each prime p, let n(p) denote the least quadratic nonresidue modulo p. Vinogradov conjectured that n(p) = O(pε) for every fixed ε > 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 2 3 (when the conductor is not cube-free) or 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 τ3 or τ4 at level above 2 3 or 3 4, respectively. Some applications to the least primitive root are also given.

quadratic nonresidue, Elliott–Halberstam conjecture, character sums, Burgess bound
Mathematical Subject Classification 2010
Primary: 11L40
Secondary: 11L20
Received: 26 October 2014
Revised: 12 January 2015
Accepted: 18 March 2015
Published: 30 May 2015
Terence Tao
Department of Mathematics
University of California, Los Angeles
405 Hilgard Avenue
Los Angeles, CA 90095-1555
United States