Journal of Integer Sequences, Vol. 6 (2003), Article 03.2.6

A Multidimensional Version of a Result of Davenport-Erdös


O-Yeat Chan, Geumlan Choi, and Alexandru Zaharescu
Department of Mathematics
University of Illinois
1409 West Green Street
Urbana, IL 61801
USA

Abstract: Davenport and Erdös showed that the distribution of values of sums of the form

\begin{displaymath} S_h(x)=\sum_{m=x+1}^{x+h} \left(\frac mp \right), \end{displaymath}

where $p$ is a prime and $\left(\frac mp \right)$ is the Legendre symbol, is normal as $h, p\to\infty$ such that $\frac{\log h}{\log p}\to 0$. We prove a similar result for sums of the form

\begin{displaymath} S_h(x_1, \ldots, x_n)=\sum_{z_1=x_1+1}^{x_1+h} \cdots \sum_{z_n=x_n+1}^{x_n+h} \left( \frac {z_1+ \cdots+z_n}{p} \right). \end{displaymath}

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Received February 24, 2002; revised version received June 16, 2003. Published in Journal of Integer Sequences July 9, 2003.

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