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

Journal article

We show that for any fixed ε > 0, there are numbers δ > 0 and p0 ≥ 2 with the following property: for every prime p ≥ p0 and every integer N such that p1/(4√e) +ε ≤ N ≤ p, the sequence 1, 2, ..., N contains at least δ N 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. © 2008 London Mathematical Society.

Authors: Banks, W; Garaev, M; Heath-Brown, D; Shparlinski, I

• Publication status: Published
• Journal: BULLETIN OF THE LONDON MATHEMATICAL SOCIETY
• Volume: 40
• Issue: 1
• Pages: 88-96
• Publication date: 2008-02-01
• DOI: 10.1112/blms/bdm111
• EISSN: 1469-2120
• ISSN: 0024-6093
• Language: English