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Long and Short Sums of a
Twisted Divisor Function Olivier Bordellès
2, allée de la Combe
43000 Aiguilhe
France
mailto:borde43@wanadoo.frborde43@wanadoo.fr

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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.