# When the number of divisors is a quadratic residue

## TL;DR

This paper investigates the behavior of a multiplicative function related to the divisor function modulo a prime, revealing dependence on quadratic residues and providing bounds for specific cases.

## Contribution

It introduces a new comparison between the mean value of a divisor-related function and the classical divisor function, highlighting the influence of quadratic residue properties.

## Key findings

- The mean value of the function depends heavily on whether 2 is a quadratic residue modulo q.
- For q=5, a bound for short sums is established using advanced techniques from the geometry of numbers.
- The results connect divisor functions, quadratic residues, and bounds on exponential sums.

## 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 $\left( \lambda_q \star \mathbf{1} \right) (n)$ 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 $\lambda_q \star \mathbf{1}$ 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.