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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
DOI: https://doi.org/10.1090/S0002-9939-2015-12584-1
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 $ 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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  • [1] D. A. Burgess, The distribution of quadratic residues and non-residues, Mathematika 4 (1957), 106-112. MR 0093504 (20 #28)
  • [2] Rainer Dietmann, Christian Elsholtz, and Igor E. Shparlinski, On gaps between quadratic non-residues in the Euclidean and Hamming metrics, Indag. Math. (N.S.) 24 (2013), no. 4, 930-938. MR 3124809, https://doi.org/10.1016/j.indag.2013.02.005
  • [3] Pál Erdős, Remarks on number theory. I, Mat. Lapok 12 (1961), 10-17 (Hungarian, with Russian and English summaries). MR 0144869 (26 #2410)
  • [4] Michael Filaseta and Ognian Trifonov, On gaps between squarefree numbers. II, J. London Math. Soc. (2) 45 (1992), no. 2, 215-221. MR 1171549 (93h:11103), https://doi.org/10.1112/jlms/s2-45.2.215
  • [5] M. Z. Garaev, S. V. Konyagin, and Yu. V. Malykhin, Asymptotics of the sum of powers of distances between power residues modulo a prime, Tr. Mat. Inst. Steklova 276 (2012), no. Teoriya Chisel, Algebra i Analiz, 83-95 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 276 (2012), no. 1, 77-89. MR 2986111, https://doi.org/10.1134/S0081543812010075
  • [6] S. W. Graham and C. J. Ringrose, Lower bounds for least quadratic nonresidues, Analytic number theory (Allerton Park, IL, 1989) Progr. Math., vol. 85, Birkhäuser Boston, Boston, MA, 1990, pp. 269-309. MR 1084186 (92d:11108)
  • [7] Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR 2061214 (2005h:11005)
  • [8] Hugh L. Montgomery and Robert C. Vaughan, Multiplicative number theory. I. Classical theory, Cambridge Studies in Advanced Mathematics, vol. 97, Cambridge University Press, Cambridge, 2007. MR 2378655 (2009b:11001)
  • [9] Gérald Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge Studies in Advanced Mathematics, vol. 46, Cambridge University Press, Cambridge, 1995. Translated from the second French edition (1995) by C. B. Thomas. MR 1342300 (97e:11005b)
  • [10] D. I. Tolev, On the distribution of $ r$-tuples of squarefree numbers in short intervals, Int. J. Number Theory 2 (2006), no. 2, 225-234. MR 2240227 (2008a:11111), https://doi.org/10.1142/S179304210600053X

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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

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