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Mathematics of Computation

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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
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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.


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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