Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-25T13:23:41.826Z Has data issue: false hasContentIssue false
  • English
  • Français

The distribution of short character sums

Published online by Cambridge University Press:  17 May 2013

YOUNESS LAMZOURI
YOUNESS LAMZOURI*
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL, 61801U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

Let χ be a non-real Dirichlet character modulo a prime q. In this paper we prove that the distribution of the short character sum Sχ,H(x) = ∑x<n≤x+H χ(n), as x runs over the positive integers below q, converges to a two-dimensional Gaussian distribution on the complex plane, provided that log H=o(log q) and H → ∞ as q → ∞. Furthermore, we use an idea of Selberg to establish an upper bound on the rate of convergence.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 155 , Issue 2 , September 2013 , pp. 207 - 218
Copyright
Copyright © Cambridge Philosophical Society 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Chatterjee, S. and Soundararajan, K.Random multiplicative functions in short intervals. Int. Math. Res. Not. (2012), no.3, 479–492.CrossRefGoogle Scholar
[2]Davenport, H. and Erdös, P.The distribution of quadratic and higher residues. Publ. Math. Debrecen 2 (1952), 252–265.CrossRefGoogle Scholar
[3]Iwaniec, H. and Kowalski, E.Analytic number theory. American Mathematical Society Colloquium Publications, 53. (Amer. Math. Soc., 2004).Google Scholar
[4]Mak, K. H. and Zaharescu, A.The distribution of values of short hybrid exponential sums on curves over finite fields. Math. Res. Lett. 18 (2011), no. 1, 155–174.CrossRefGoogle Scholar
[5]Ng, N.The Möbius function in short intervals. Anatomy of Integers, CRM Proceedings and Lecture Notes, Vol. 46, (2008), 247–258.CrossRefGoogle Scholar
[6]Tsang, K. M. The distribution of the values of the zeta function. Thesis (Princeton University, October 1984), 179 pp.Google Scholar
[7]Selberg, A.Old and new conjectures and results about a class of Dirichlet series. Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989) Univ. Salerno, Salerno (1992), 367–385.Google Scholar
[8]Weil, A.On some exponential sums. Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 204–207.CrossRefGoogle ScholarPubMed