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\begin{center}
\vskip 1cm{\LARGE\bf 
Long and Short Sums of a \\
\vskip .1in
Twisted Divisor Function}
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\large
Olivier Bordell\`es\\
2, all\'{e}e de la Combe\\
43000 Aiguilhe\\
France\\
\href{mailto:borde43@wanadoo.fr}{\tt borde43@wanadoo.fr}
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\begin{abstract}
Let $q > 2$ be a prime number and define $\lambda_q := \left(
\frac{\tau}{q} \right)$ where $\tau(n)$ is the number of divisors of
$n$ and $\left( \frac{\cdot}{q} \right)$ is the Legendre symbol. When
$\tau(n)$ is a quadratic residue modulo $q$, then the convolution
$\left( \lambda_q \star \mathbf{1} \right) (n)$ could be close to the
number of divisors of $n$. The aim of this work is to compare the mean
value of the function $\lambda_q \star \mathbf{1}$ to the well known
average order of $\tau$. A bound for short sums in the case $q=5$ is
also given, using profound results from the theory of integer points
close to certain smooth curves.
\end{abstract}


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