Journal of the American Mathematical Society

Author(s): Andrew Granville; K. Soundararajan
Journal: J. Amer. Math. Soc. 20 (2007), 357-384.
MSC (2000): Primary 11L40
Posted: May 26, 2006
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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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Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 11L40

Andrew Granville
Affiliation: D{é}partement de Math{é}matiques et Statistique, Universit{é} de Montr{é}al, CP 6128 succ Centre-Ville, Montr{é}al, 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: 10.1090/S0894-0347-06-00536-4
PII: S 0894-0347(06)00536-4
Received by editor(s): March 2, 2005
Posted: 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.
Copyright of article: Copyright 2006, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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ISSN: 1088-6834(e) ISSN: 0894-0347(p)

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