Mean squares of quadratic twists of the M\"obius function

TL;DR

This paper provides an asymptotic evaluation of a sum involving quadratic twists of the Möbius function, revealing new insights into their distribution over square-free integers.

Contribution

It offers the first asymptotic formula for the mean squares of quadratic twists of the Möbius function over square-free integers.

Findings

Asymptotic formula derived for the sum involving quadratic twists

Insights into the distribution of quadratic twists of the Möbius function

Advances understanding of the behavior of the Möbius function in quadratic settings

Abstract

In this paper, we evaluate asymptotically the sum \[ \sum_{d \leq X} \left( \sum_{n \leq Y} \mu(n)\leg {8d}{n} \right)^2, \] where $\leg {8d}{n}$ is the Kronecker symbol and $d$ runs over positive, odd, square-free integers.