Density of non-residues in Burgess-type intervals and applications

2006/07/31 by W. D. Banks, M. Z. Garaev, D. R. Heath-Brown +1 · 1 citation
Mathematics · #math.NT #msc:11N69

paper · pdf · doi:10.1112/blms/bdm111

In the new version we use an idea of Roger Heath-Brown (who is now a co-author) to simply the proof and improve the main results of the previous version, 14 pages

arxiv created 2007/09/25 · arxiv updated 2014/02/26

Abstract

We show that for any fixed \eps>0, there are numbers δ>0 and p₀≥ 2 with the following property: for every prime p≥ p₀ and every integer N such that p^{1/(4√(e))+\eps}≤ N≤ p, the sequence 1,2,...,N contains at least δN quadratic non-residues modulo p. We use this result to obtain strong upper bounds on the sizes of the least quadratic non-residues in Beatty and Piatetski--Shapiro sequences.

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