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