Higher Moments for Polynomial Chowla

Abstract Let $\lambda (n)$ be the Liouville function. We study the distribution of $$ \begin{align*} & \frac{1}{x^{1/2}}\sum_{x\leqslant n\leqslant 2x}\lambda(f(n)) \end{align*} $$ 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.