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THE DISTRIBUTION OF THE MAXIMUM OF CHARACTER SUMS

Part of: Exponential sums and character sums

Published online by Cambridge University Press:  08 January 2013

Jonathan W. Bober  and
Leo Goldmakher
Jonathan W. Bober
Affiliation:
Department of Mathematics, University of Washington, Seattle, WA, U.S.A. (email: jwbober@math.washington.edu)
Leo Goldmakher
Affiliation:
Department of Mathematics, University of Toronto, Toronto, ON, Canada (email: leo.goldmakher@utoronto.ca)
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Abstract

We obtain explicit bounds on the moments of character sums, refining estimates of Montgomery and Vaughan. As an application we obtain results on the distribution of the maximal magnitude of character sums normalized by the square root of the modulus, finding almost double exponential decay in the tail of this distribution.

MSC classification

Secondary: 11L40: Estimates on character sums
Type
Research Article
Information
Mathematika , Volume 59 , Issue 2 , July 2013 , pp. 427 - 442
Copyright
Copyright © 2013 University College London 

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References

[1]Gallagher, P. X. and Montgomery, H. L., A note on Burgess’s estimate. Math. Notes 88 (2010), 321–329.CrossRefGoogle Scholar
[2]Goldmakher, L., Multiplicative mimicry and improvements of the Pólya–Vinogradov inequality. Algebra Number Theory 6 (2012), 123–163.CrossRefGoogle Scholar
[3]Goldmakher, L., Character sums to smooth moduli are small. Canad. J. Math. 62 (2010), 1099–1115.CrossRefGoogle Scholar
[4]Graham, S. W. and Ringrose, C. J., Lower bounds for least quadratic nonresidues. In Analytic Number Theory (Allerton Park, Illinois, 1989) (Progress in Mathematics 85), Birkhäuser (Boston, MA, 1990), 269–309.CrossRefGoogle Scholar
[5]Granville, A. and Soundararajan, K., Large character sums: pretentious characters and the Pólya Vinogradov theorem. J. Amer. Math. Soc. 20(2) (2007), 357–384.CrossRefGoogle Scholar
[6]Granville, A. and Soundararajan, K., Extreme values of $|\zeta (1 + it)|$. In The Riemann Zeta Function and Related Themes: Papers in Honour of Professor K. Ramachandra (Ramanujan Mathematical Society Lecture Notes Series 2), Ramanujan Mathematical Society (Mysore, 2006), 65–80.Google Scholar
[7]Iwaniec, H., On zeros of Dirichlet’s $L$ series. Invent. Math. 23 (1974), 97–104.CrossRefGoogle Scholar
[8]Montgomery, H. L. and Vaughan, R. C., Mean values of character sums. Canad. J. Math. 31(3) (1979), 476–487.CrossRefGoogle Scholar
[9]Montgomery, H. L. and Vaughan, R. C., Exponential sums with multiplicative coefficients. Invent. Math. 43(1) (1977), 69–82.CrossRefGoogle Scholar
[10]Norton, K. K., Upper bounds for sums of powers of divisor functions. J. Number Theory 40 (1992), 60–85.CrossRefGoogle Scholar
[11]Paley, R. E. A. C., A theorem on characters. J. Lond. Math. Soc. 7(1) (1932), 28–32.CrossRefGoogle Scholar