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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.
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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 

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References

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