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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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Large character sums: Pretentious characters and the Pólya-Vinogradov theorem
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by Andrew Granville and K. Soundararajan
J. Amer. Math. Soc. 20 (2007), 357-384
Published electronically: May 26, 2006


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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Bibliographic Information
  • Andrew Granville
  • Affiliation: Département de Mathématiques et Statistique, Université de Montréal, CP 6128 succ Centre-Ville, Montréal, Quebec H3C 3J7, Canada
  • MR Author ID: 76180
  • ORCID: 0000-0001-8088-1247
  • Email:
  • 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
  • MR Author ID: 319775
  • Email:
  • 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.
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 20 (2007), 357-384
  • MSC (2000): Primary 11L40
  • DOI:
  • MathSciNet review: 2276774