Infinitely many sign changes of the Liouville function on $x^2+d$

TL;DR

This paper proves that the Liouville function, which indicates the parity of the number of prime factors, changes sign infinitely often along the quadratic polynomial sequence x^2+d.

Contribution

It establishes the infinite sign change of the Liouville function specifically on quadratic polynomial sequences, a significant step in understanding its oscillatory behavior.

Findings

Liouville function changes sign infinitely often on x^2+d

Supports conjectures about sign oscillations in polynomial sequences

Advances understanding of multiplicative function behavior on quadratic forms

Abstract

We show that the Liouville function changes sign infinitely often for $x^2+d$.