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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

Large character sums: Pretentious characters and the Pólya-Vinogradov theorem


Authors: Andrew Granville and K. Soundararajan
Journal: J. Amer. Math. Soc. 20 (2007), 357-384
MSC (2000): Primary 11L40
Published electronically: May 26, 2006
MathSciNet review: 2276774
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Abstract: In 1918 Pólya and Vinogradov gave an upper bound for the maximal size of character sums, which still remains the best known general estimate. One of the main results of this paper provides a substantial improvement of the Pólya-Vinogradov bound for characters of odd, bounded order. In 1977 Montgomery and Vaughan showed how the Pólya-Vinogradov inequality may be sharpened assuming the Generalized Riemann Hypothesis. We give a simple proof of their estimate and provide an improvement for characters of odd, bounded order. The paper also gives characterizations of the characters for which the maximal character sum is large, and it finds a hidden structure among these characters.


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Additional Information

Andrew Granville
Affiliation: Département de Mathématiques et Statistique, Université de Montréal, CP 6128 succ Centre-Ville, Montréal, Quebec H3C 3J7, Canada
Email: andrew@dms.umontreal.ca

K. Soundararajan
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Address at time of publication: Department of Mathematics, Stanford University, Building 380, 450 Serra Mall, Stanford, California 94305-2125
Email: ksound@umich.edu

DOI: http://dx.doi.org/10.1090/S0894-0347-06-00536-4
PII: S 0894-0347(06)00536-4
Received by editor(s): March 2, 2005
Published electronically: May 26, 2006
Additional Notes: Le premier auteur est partiellement soutenu par une bourse de la Conseil de recherches en sciences naturelles et engénie du Canada.
The second author is partially supported by the National Science Foundation.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.