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
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
F.R.K. Chung, R.L. Graham, and R.M. Wilson. Quasi-random graphs, preprint.
J. G. van der Coiput, Verschärfung der Abschätzungen beim Teilerprob- lem, Math. Annalen 89 (1922), 39–65.
H. Davenport, Multiplicative Number Theory (2nd edition, revised by H.L. Montgomery), GTM 74, Springer-Verlag, Berlin- Heidelberg-New York, 1980.
V.R. Fridlender, On the least n-th power non-residue, Dokl. Akad. Nauk SSSR 66 (1949), 351–352.
S. W. Graham, An asymptotic formula related to the Selberg sieve, J. Number Theory 10 (1978), 83–94.
S. W. Graham, On Linnik’s constant, Acta Arith. 39 (1980), 163–179.
H. Halberstam and H.-E. Richert, Sieve Methods, Academic Press, London, 1974.
D. R. Heath-Brown, Hybrid bounds for Dirichlet L-functions, Invent. Math. 47 (1978), 148–170.
C. Hooley, On the Brun-Titchmarsh theorem, J. Reine Angew. Math. 255 (1972), 60–79.
A. Ivic, The Riemann zeta-function, Wiley-Interscience, New York, 1985.
M. Jutila, On Linnik’s Constant, Math. Scand. 41 (1977), 45–62.
H. Maier, Chains of large gaps between consecutive primes, Adv. in Math. 39 (3) (1981), 257–269.
H.L. Montgomery, Topics in Multiplicative Number Theory, Lecture Notes in Math. 227, Springer-Verlag, New York, 1971.
C. Ringrose, The q-analogue of van der Corput’s method, Thesis, University of Oxford, Oxford, 1985.
H. Salié, Uber den kleinsten positiven quadratischen Nichtrest nach einer Primzahl, Math. Nachr. 3 (1949), 7–8.
W. Schmidt, Equations over finite fields: an elementary approach, Lec-ture Notes in Math. 536, Springer-Verlag, New York, 1976.
B. R. Srinivasan, The lattice point problem of many-dimensional hyper- boloids II, Acta Arith. 8 (1963) 173–204.
E. C. Titchmarsh, The theory of the Riemann zeta-function, (2nd edition, revised by D.R. Heath-Brown) Clarendon Press, Oxford, 1986.
A. Weil, On some exponential sums, Proc. Nat. Acad. Sci. U.S.A., 34 (1948) 204–207.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Additional information
Dedicated to Paul Bateman
Rights 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