Higher Moments for Polynomial Chowla

TL;DR

This paper investigates the distribution of sums involving the Liouville function evaluated at polynomial values, demonstrating that the first d+1 moments follow a Gaussian distribution for random polynomials.

Contribution

It establishes the Gaussian nature of the first d+1 moments for sums of the Liouville function over random polynomials of fixed degree.

Findings

First d+1 moments are Gaussian.

Distribution of sums approximates a normal distribution.

Results hold for polynomials with bounded coefficients.

Abstract

Let $\lambda(n)$ be the Liouville function. We study the distribution of \[ \frac{1}{x^{1/2}}\sum_{x\leq n\leq 2x}\lambda(f(n)) \] over random polynomials $f$ of fixed degree $d$ and coefficients bounded in magnitude by $H$. In particular we prove that the first $d+1$ moments are Gaussian.