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Density of non‐residues in Burgess‐type intervals and applications
Author(s) -
Banks W. D.,
Garaev M. Z.,
HeathBrown D. R.,
Shparlinski I. E.
Publication year - 2008
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdm111
Subject(s) - combinatorics , modulo , quadratic residue , mathematics , integer (computer science) , sequence (biology) , quadratic equation , type (biology) , prime (order theory) , upper and lower bounds , chemistry , geometry , mathematical analysis , computer science , biology , ecology , biochemistry , programming language
We show that for any fixed ε > 0, there are numbers δ > 0 and p 0 ⩾ 2 with the following property: for every prime p ⩾ p 0 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.