Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Conditional bounds for the least quadratic non-residue and related problems


Authors: Youness Lamzouri, Xiannan Li and Kannan Soundararajan
Journal: Math. Comp. 84 (2015), 2391-2412
MSC (2010): Primary 11N60; Secondary 11R42
DOI: https://doi.org/10.1090/S0025-5718-2015-02925-1
Published electronically: January 26, 2015
Corrigendum: Math. Comp. 86 (2017), 2551-2554.
MathSciNet review: 3356031
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper studies explicit and theoretical bounds for several interesting quantities in number theory, conditionally on the Generalized Riemann Hypothesis. Specifically, we improve the existing explicit bounds for the least quadratic non-residue and the least prime in an arithmetic progression. We also refine the classical conditional bounds of Littlewood for $ L$-functions at $ s=1$. In particular, we derive explicit upper and lower bounds for $ L(1,\chi )$ and $ \zeta (1+it)$, and deduce explicit bounds for the class number of imaginary quadratic fields. Finally, we improve the best known theoretical bounds for the least quadratic non-residue, and more generally, the least $ k$-th power non-residue.


References [Enhancements On Off] (What's this?)

  • [1] N. C. Ankeny, The least quadratic non residue, Ann. of Math. (2) 55 (1952), 65-72. MR 0045159 (13,538c)
  • [2] Eric Bach, Explicit bounds for primality testing and related problems, Math. Comp. 55 (1990), no. 191, 355-380. MR 1023756 (91m:11096), https://doi.org/10.2307/2008811
  • [3] Andrew R. Booker, Quadratic class numbers and character sums, Math. Comp. 75 (2006), no. 255, 1481-1492 (electronic). MR 2219039 (2008a:11140), https://doi.org/10.1090/S0025-5718-06-01850-3
  • [4] Eric Bach and Jonathan Sorenson, Explicit bounds for primes in residue classes, Math. Comp. 65 (1996), no. 216, 1717-1735. MR 1355006 (97a:11143), https://doi.org/10.1090/S0025-5718-96-00763-6
  • [5] Emanuel Carneiro and Vorrapan Chandee, Bounding $ \zeta (s)$ in the critical strip, J. Number Theory 131 (2011), no. 3, 363-384. MR 2739041 (2011k:11109), https://doi.org/10.1016/j.jnt.2010.08.002
  • [6] Vorrapan Chandee, Explicit upper bounds for $ L$-functions on the critical line, Proc. Amer. Math. Soc. 137 (2009), no. 12, 4049-4063. MR 2538566 (2010i:11134), https://doi.org/10.1090/S0002-9939-09-10075-8
  • [7] Vorrapan Chandee and K. Soundararajan, Bounding $ \vert \zeta (\frac 12+it)\vert $ on the Riemann hypothesis, Bull. Lond. Math. Soc. 43 (2011), no. 2, 243-250. MR 2781205 (2012c:11172), https://doi.org/10.1112/blms/bdq095
  • [8] Harold Davenport, Multiplicative Number Theory, 3rd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York, 2000. Revised and with a preface by Hugh L. Montgomery. MR 1790423 (2001f:11001)
  • [9] James L. Hafner and Kevin S. McCurley, A rigorous subexponential algorithm for computation of class groups, J. Amer. Math. Soc. 2 (1989), no. 4, 837-850. MR 1002631 (91f:11090), https://doi.org/10.2307/1990896
  • [10] D. R. Heath-Brown, Zero-free regions for Dirichlet $ L$-functions, and the least prime in an arithmetic progression, Proc. London Math. Soc. (3) 64 (1992), no. 2, 265-338. MR 1143227 (93a:11075), https://doi.org/10.1112/plms/s3-64.2.265
  • [11] 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)
  • [12] Michael J. Jacobson Jr., Shantha Ramachandran, and Hugh C. Williams, Numerical results on class groups of imaginary quadratic fields, Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 4076, Springer, Berlin, 2006, pp. 87-101. MR 2282917 (2007j:11178), https://doi.org/10.1007/11792086_7
  • [13] J. E. Littlewood, On the zeros of the Riemann zeta-function, Math. Proc. Cambridge Philos. Soc. 22 (1924) 295-318.
  • [14] J. E. Littlewood, On the function $ 1/\zeta (1+it)$, Proc. London Math. Soc. 27 (1928) 349-357.
  • [15] Mark Watkins, Class numbers of imaginary quadratic fields, Math. Comp. 73 (2004), no. 246, 907-938 (electronic). MR 2031415 (2005a:11175), https://doi.org/10.1090/S0025-5718-03-01517-5
  • [16] S. Wedeniwski, Primality Tests on Commutator Curves. Dissertation: Tübingen, Germany, 2001.
  • [17] Triantafyllos Xylouris, On the least prime in an arithmetic progression and estimates for the zeros of Dirichlet $ L$-functions, Acta Arith. 150 (2011), no. 1, 65-91. MR 2825574 (2012m:11129), https://doi.org/10.4064/aa150-1-4

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 11N60, 11R42

Retrieve articles in all journals with MSC (2010): 11N60, 11R42


Additional Information

Youness Lamzouri
Affiliation: Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON, M3J1P3
Email: lamzouri@mathstat.yorku.ca

Xiannan Li
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, Illinois 61801
Email: xiannan@illinois.edu

Kannan Soundararajan
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
Email: ksound@math.stanford.edu

DOI: https://doi.org/10.1090/S0025-5718-2015-02925-1
Received by editor(s): September 15, 2013
Received by editor(s) in revised form: November 25, 2013
Published electronically: January 26, 2015
Additional Notes: The first author was supported in part by an NSERC Discovery grant. The third author was supported in part by NSF grant DMS-1001068, and a Simons Investigator grant from the Simons Foundation
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society