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
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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.
- P. T. Bateman and S. Chowla, Averages of character sums, Proc. Amer. Math. Soc. 1 (1950), 781–787. MR 42445, DOI https://doi.org/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 https://doi.org/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
- 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
- Andrew Granville and K. Soundararajan, Large character sums, J. Amer. Math. Soc. 14 (2001), no. 2, 365–397. MR 1815216, DOI https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/10.1016/0022-314X%2888%2990106-0 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 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.
Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 11L40
Retrieve articles in all journals with MSC (2000): 11L40
Additional 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.
Article copyright:
© Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
