When the number of divisors is a quadratic residue

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 $\left({\lambda }_q\star 1\right)(n)$ could be close to the number of divisors of $n$. This is the aim of this work to compare the mean value of the function ${\lambda }_q\star 1$ to the well known average order of $\tau$. The proof reveals that the results depend heavily on the value of $\left(\frac{2}{q}\right)$. 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.

9 pages

References in corpus (1)

• Notes de lecture de l'article "Partial sums of the Möbius function" de Kannan Soundararajan