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Davenport and Erd\H{o}s 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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