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

W. D. Banks; M. Z. Garaev; D. R. Heath-Brown; I. E. Shparlinski

doi:10.1112/blms/bdm111

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

W. D. Banks, M. Z. Garaev, D. R. Heath-Brown, I. E. Shparlinski

School of Computing

Research output: Contribution to journal › Article › peer-review

25 Citations (Scopus)

Abstract

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 p^{1/(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.

Original language	English
Pages (from-to)	88-96
Number of pages	9
Journal	Bulletin of the London Mathematical Society
Volume	40
Issue number	1
DOIs	https://doi.org/10.1112/blms/bdm111
Publication status	Published - Feb 2008

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

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