Long and Short Sums of a Twisted Divisor Function

Let

be a prime number and define $\lambda_q := \left( \frac{\tau}{q} \right)$ where $\tau(n)$ is the number of divisors of

and $\left( \frac{\cdot}{q} \right)$ is the Legendre symbol. When $\tau(n)$ is a quadratic residue modulo

, then the convolution $\left( \lambda_q \star \mathbf{1} \right) (n)$ could be close to the number of divisors of

. 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

is also given, using profound results from the theory of integer points close to certain smooth curves.

Long and Short Sums of a Twisted Divisor Function

Olivier Bordellès 2, allée de la Combe 43000 Aiguilhe France

Olivier Bordellès
2, allée de la Combe
43000 Aiguilhe
France