On the moments of averages of quadratic twists of the M\"obius function

TL;DR

This paper studies the statistical moments of quadratic twists of the Möbius function, providing unconditional asymptotic results for their behavior, which advances understanding of their distribution and properties.

Contribution

It introduces a new analysis of moments of quadratic twists of the Möbius function, offering the first unconditional asymptotic formulas for these sums.

Findings

Derived asymptotic formulas for moments of quadratic twists of the Möbius function.

Established unconditional results for the behavior of these moments.

Enhanced understanding of the distribution of quadratic twists of the Möbius function.

Abstract

We consider the moment of quadratic twists of the M\"obius function of the form \[ S_k(X,Y) = \sum_{d\leq X} \left( \sum_{n\leq Y} \left(\frac{8d}{n}\right) \mu(n)\right)^k, \] where $\left(\frac{8d}{\cdot}\right)$ is the Kronecker symbol and $d$ runs over positive, odd and square-free integers. We give unconditional results for their asymptotic behaviors.