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

Granville, Andrew; Soundararajan, K.

doi:10.1090/S0894-0347-06-00536-4

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
DOI: https://doi.org/10.1090/S0894-0347-06-00536-4
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.

1. P. T. Bateman and S. Chowla, Averages of character sums, Proc. Amer. Math. Soc. 1 (1950), 781–787. MR 42445, https://doi.org/10.1090/S0002-9939-1950-0042445-6
2. D. A. Burgess, On character sums and 𝐿-series, Proc. London Math. Soc. (3) 12 (1962), 193–206. MR 0132733, https://doi.org/10.1112/plms/s3-12.1.193
D. A. Burgess, On character sums and 𝐿-series. II, Proc. London Math. Soc. (3) 13 (1963), 524–536. MR 0148626, https://doi.org/10.1112/plms/s3-13.1.524
3. S. Chowla, On the class-number of the corpus 𝑃(√-𝑘), Proc. Nat. Inst. Sci. India 13 (1947), 197–200. MR 27303
4. 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
5. 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
6. Andrew Granville and K. Soundararajan, Large character sums, J. Amer. Math. Soc. 14 (2001), no. 2, 365–397. MR 1815216, https://doi.org/10.1090/S0894-0347-00-00357-X
7. Andrew Granville and K. Soundararajan, The spectrum of multiplicative functions, Ann. of Math. (2) 153 (2001), no. 2, 407–470. MR 1829755, https://doi.org/10.2307/2661346
8. A. Granville and K. Soundararajan, The distribution of values of 𝐿(1,𝜒_{𝑑}), Geom. Funct. Anal. 13 (2003), no. 5, 992–1028. MR 2024414, https://doi.org/10.1007/s00039-003-0438-3
9. Andrew Granville and K. Soundararajan, Upper bounds for |𝐿(1,𝜒)|, Q. J. Math. 53 (2002), no. 3, 265–284. MR 1930263, https://doi.org/10.1093/qjmath/53.3.265
10. Adolf Hildebrand, Large values of character sums, J. Number Theory 29 (1988), no. 3, 271–296. MR 955953, https://doi.org/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.
12. H. L. Montgomery and R. C. Vaughan, Exponential sums with multiplicative coefficients, Invent. Math. 43 (1977), no. 1, 69–82. MR 457371, https://doi.org/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.

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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: https://doi.org/10.1090/S0894-0347-06-00536-4
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.