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 | References | Similar Articles | Additional Information

Abstract: We use the Burgess bound and combinatorial sieve to obtain an upper bound on the number of primes in a dyadic interval for which a given interval does not contain a quadratic non-residue modulo . The bound is non-trivial for any function as . This is an analogue of the well-known estimates on the smallest quadratic non-residue modulo on average over primes , which corresponds to the choice .

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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