Sign Changes of the Liouville function on quadratics

TL;DR

This paper investigates the sign-changing behavior of the Liouville function on quadratic polynomials, providing conditions under which the function changes sign infinitely often, thus contributing to the understanding of Chowla's conjecture for degree two.

Contribution

The paper proves that for certain quadratic polynomials, the Liouville function changes sign infinitely often, advancing partial results towards Chowla's conjecture for degree two polynomials.

Findings

Proves infinite sign changes of the Liouville function for specific quadratic polynomials.

Establishes a link between solutions to quadratic equations and sign changes of the Liouville function.

Provides partial progress on Chowla's conjecture for quadratic polynomials.

Abstract

Let $\lambda (n)$ denote the Liouville function. Complementary to the prime number theorem, Chowla conjectured that \vspace{1mm} \noindent {\bf Conjecture (Chowla).} {\em \begin{equation} \label{a.1} \sum_{n\le x} \lambda (f(n)) =o(x) \end{equation} for any polynomial $f(x)$ with integer coefficients which is not of form $bg(x)^2$. } \vspace{1mm} \noindent The prime number theorem is equivalent to \eqref{a.1} when $f(x)=x$. Chowla's conjecture is proved for linear functions but for the degree greater than 1, the conjecture seems to be extremely hard and still remains wide open. One can consider a weaker form of Chowla's conjecture, namely, \vspace{1mm} \noindent {\bf Conjecture 1 (Cassaigne, et al).} {\em If $f(x) \in \Z [x]$ and is not in the form of $bg^2(x)$ for some $g(x)\in \Z[x]$, then $\lambda (f(n))$ changes sign infinitely often.} Clearly, Chowla's conjecture implies Conjecture 1. Although it is weaker, Conjecture 1 is still wide open for polynomials of degree $>1$. In this article, we study Conjecture 1 for the quadratic polynomials. One of our main theorems is {\bf Theorem 1.} {\em Let $f(x) = ax^2+bx +c $ with $a>0$ and $l$ be a positive integer such that $al$ is not a perfect square. Then if the equation $f(n)=lm^2 $ has one solution $(n_0,m_0) \in \Z^2$, then it has infinitely many positive solutions $(n,m) \in \N^2$.} As a direct consequence of Theorem 1, we prove some partial results of Conjecture 1 for quadratic polynomials are also proved by using Theorem 1.