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Quadratic non-residues in short intervals


Authors: Sergei V. Konyagin and Igor E. Shparlinski
Journal: Proc. Amer. Math. Soc. 143 (2015), 4261-4269
MSC (2010): Primary 11A15, 11L40
Published electronically: March 31, 2015
MathSciNet review: 3373925
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Abstract: We use the Burgess bound and combinatorial sieve to obtain an upper bound on the number of primes $ p$ in a dyadic interval $ [Q,2Q]$ for which a given interval $ [u+1,u+\psi (Q)]$ does not contain a quadratic non-residue modulo $ p$. The bound is non-trivial for any function $ \psi (Q)\to \infty $ as $ Q\to \infty $. This is an analogue of the well-known estimates on the smallest quadratic non-residue modulo $ p$ on average over primes $ p$, which corresponds to the choice $ u=0$.


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Additional Information

Sergei V. Konyagin
Affiliation: Steklov Mathematical Institute, 8, Gubkin Street, Moscow, 119991, Russia
Email: konyagin@mi.ras.ru

Igor E. Shparlinski
Affiliation: Department of Pure Mathematics, University of New South Wales, Sydney, NSW 2052, Australia
Email: igor.shparlinski@unsw.edu.au

DOI: https://doi.org/10.1090/S0002-9939-2015-12584-1
Keywords: Quadratic non-residues, character sums
Received by editor(s): November 27, 2013
Received by editor(s) in revised form: June 21, 2014
Published electronically: March 31, 2015
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2015 American Mathematical Society