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

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- by Youness Lamzouri, Xiannan Li and Kannan Soundararajan PDF
- Math. Comp.
**84**(2015), 2391-2412 Request permission

## 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

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## Additional Information

**Youness Lamzouri**- Affiliation: Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON, M3J1P3
- MR Author ID: 804642
- Email: lamzouri@mathstat.yorku.ca
**Xiannan Li**- Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, Illinois 61801
- MR Author ID: 867056
- Email: xiannan@illinois.edu
**Kannan Soundararajan**- Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
- MR Author ID: 319775
- Email: ksound@math.stanford.edu
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Math. Comp.
**84**(2015), 2391-2412 - MSC (2010): Primary 11N60; Secondary 11R42
- DOI: https://doi.org/10.1090/S0025-5718-2015-02925-1
- MathSciNet review: 3356031