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
- DOI: https://doi.org/10.1090/S0894-0347-06-00536-4
- Published electronically: 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.References
- P. T. Bateman and S. Chowla, Averages of character sums, Proc. Amer. Math. Soc. 1 (1950), 781–787. MR 42445, DOI 10.1090/S0002-9939-1950-0042445-6
- D. A. Burgess, On character sums and $L$-series, Proc. London Math. Soc. (3) 12 (1962), 193–206. MR 132733, DOI 10.1112/plms/s3-12.1.193
- S. Chowla, On the class-number of the corpus $P(\sqrt {-k})$, Proc. Nat. Inst. Sci. India 13 (1947), 197–200. MR 27303
- Harold Davenport, Multiplicative number theory, 2nd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York-Berlin, 1980. Revised by Hugh L. Montgomery. MR 606931, DOI 10.1007/978-1-4757-5927-3
- P. Deligne, Cohomologie étale, Lecture Notes in Mathematics, vol. 569, Springer-Verlag, Berlin, 1977 (French). Séminaire de géométrie algébrique du Bois-Marie SGA $4\frac {1}{2}$. MR 463174, DOI 10.1007/BFb0091526
- Andrew Granville and K. Soundararajan, Large character sums, J. Amer. Math. Soc. 14 (2001), no. 2, 365–397. MR 1815216, DOI 10.1090/S0894-0347-00-00357-X
- Andrew Granville and K. Soundararajan, The spectrum of multiplicative functions, Ann. of Math. (2) 153 (2001), no. 2, 407–470. MR 1829755, DOI 10.2307/2661346
- A. Granville and K. Soundararajan, The distribution of values of $L(1,\chi _d)$, Geom. Funct. Anal. 13 (2003), no. 5, 992–1028. MR 2024414, DOI 10.1007/s00039-003-0438-3
- Andrew Granville and K. Soundararajan, Upper bounds for $|L(1,\chi )|$, Q. J. Math. 53 (2002), no. 3, 265–284. MR 1930263, DOI 10.1093/qjmath/53.3.265
- Adolf Hildebrand, Large values of character sums, J. Number Theory 29 (1988), no. 3, 271–296. MR 955953, DOI 10.1016/0022-314X(88)90106-0 11 J.E. Littlewood, On the class number of the corpus $P(\sqrt {-k})$, Proc. London Math. Soc. 27 (1928), 358-372.
- H. L. Montgomery and R. C. Vaughan, Exponential sums with multiplicative coefficients, Invent. Math. 43 (1977), no. 1, 69–82. MR 457371, DOI 10.1007/BF01390204 13 R.E.A.C. Paley, A theorem on characters, J. London Math. Soc. 7 (1932), 28–32. 14 G. Pólya, Über die Verteilung der quadratischen Reste und Nichtreste, Göttingen Nachrichten (1918), 21-29. 15 I.M. Vinogradov, Über die Verteilung der quadratischen Reste und Nichtreste, J. Soc. Phys. Math. Univ. Permi 2 (1919), 1-14.
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: 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
- MR Author ID: 319775
- Email: ksound@umich.edu
- 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: https://doi.org/10.1090/S0894-0347-06-00536-4
- MathSciNet review: 2276774