Skip to main content
SpringerLink
Log in

Lower Bounds for Least Quadratic Non-Residues

  • Chapter
Analytic Number Theory

Part of the book series: Progress in Mathematics ((PM,volume 85))

Abstract

Let p be a prime, and let n p denote the least positive integer n such that n is a quadratic non-residue mod p. In 1949, Fridlender [F] and Salié [Sa] independently showed that \( {n_p} = \Omega \left( {\log p} \right) \); in other words, there are infinitely many primes p such that \( {n_p} \geqslant c\log p \) for some absolute constant c. In 1971, Montgomery showed that if the Generalized Riemann Hypothesis is true, then

$$ {n_p} = \Omega \left( {\log p\log \log p} \right) $$
(1.1)

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. F.R.K. Chung, R.L. Graham, and R.M. Wilson. Quasi-random graphs, preprint.

    Google Scholar 

  2. J. G. van der Coiput, Verschärfung der Abschätzungen beim Teilerprob- lem, Math. Annalen 89 (1922), 39–65.

    Google Scholar 

  3. H. Davenport, Multiplicative Number Theory (2nd edition, revised by H.L. Montgomery), GTM 74, Springer-Verlag, Berlin- Heidelberg-New York, 1980.

    Google Scholar 

  4. V.R. Fridlender, On the least n-th power non-residue, Dokl. Akad. Nauk SSSR 66 (1949), 351–352.

    MathSciNet  MATH  Google Scholar 

  5. S. W. Graham, An asymptotic formula related to the Selberg sieve, J. Number Theory 10 (1978), 83–94.

    Article  MathSciNet  MATH  Google Scholar 

  6. S. W. Graham, On Linnik’s constant, Acta Arith. 39 (1980), 163–179.

    Google Scholar 

  7. H. Halberstam and H.-E. Richert, Sieve Methods, Academic Press, London, 1974.

    MATH  Google Scholar 

  8. D. R. Heath-Brown, Hybrid bounds for Dirichlet L-functions, Invent. Math. 47 (1978), 148–170.

    Article  MathSciNet  Google Scholar 

  9. C. Hooley, On the Brun-Titchmarsh theorem, J. Reine Angew. Math. 255 (1972), 60–79.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Ivic, The Riemann zeta-function, Wiley-Interscience, New York, 1985.

    MATH  Google Scholar 

  11. M. Jutila, On Linnik’s Constant, Math. Scand. 41 (1977), 45–62.

    MathSciNet  MATH  Google Scholar 

  12. H. Maier, Chains of large gaps between consecutive primes, Adv. in Math. 39 (3) (1981), 257–269.

    Article  MathSciNet  MATH  Google Scholar 

  13. H.L. Montgomery, Topics in Multiplicative Number Theory, Lecture Notes in Math. 227, Springer-Verlag, New York, 1971.

    Book  Google Scholar 

  14. C. Ringrose, The q-analogue of van der Corput’s method, Thesis, University of Oxford, Oxford, 1985.

    Google Scholar 

  15. H. Salié, Uber den kleinsten positiven quadratischen Nichtrest nach einer Primzahl, Math. Nachr. 3 (1949), 7–8.

    Article  MathSciNet  Google Scholar 

  16. W. Schmidt, Equations over finite fields: an elementary approach, Lec-ture Notes in Math. 536, Springer-Verlag, New York, 1976.

    Google Scholar 

  17. B. R. Srinivasan, The lattice point problem of many-dimensional hyper- boloids II, Acta Arith. 8 (1963) 173–204.

    MathSciNet  Google Scholar 

  18. E. C. Titchmarsh, The theory of the Riemann zeta-function, (2nd edition, revised by D.R. Heath-Brown) Clarendon Press, Oxford, 1986.

    Google Scholar 

  19. A. Weil, On some exponential sums, Proc. Nat. Acad. Sci. U.S.A., 34 (1948) 204–207.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Michigan Technological University, Houghton, MI, 49931, USA

    S. W. Graham

  2. 3 The Laurels Gledhow Lane, Leeds, LS8 1PD, England

    C. J. Ringrose

Authors
  1. S. W. Graham
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. J. Ringrose
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 west Green Street, 61801, Urbana, Illinois, USA

    Bruce C. Berndt , Harold G. Diamond , Heini Halberstam  & Adolf Hildebrand , ,  & 

Additional information

Dedicated to Paul Bateman

Rights and permissions

Reprints and permissions

Copyright information

© 1990 Birkhäuser Boston

About this chapter

Cite this chapter

Graham, S.W., Ringrose, C.J. (1990). Lower Bounds for Least Quadratic Non-Residues. In: Berndt, B.C., Diamond, H.G., Halberstam, H., Hildebrand, A. (eds) Analytic Number Theory. Progress in Mathematics, vol 85. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-3464-7_18

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-3464-7_18

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-0-8176-3481-0

  • Online ISBN: 978-1-4612-3464-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation