Abstract

Let $x_1, \ldots, x_n$ be pairwise coprime positive integers. We study the parity behavior of the Liouville function $$\lambda(x_1 x_2 \cdots x_n (x_1 + x_2 + \cdots + x_n)), \quad \lambda(k) = (-1)^{\Omega(k)}.$$ We formulate the heuristic conjecture that, under natural density assumptions on such tuples, the values $+1$ and $-1$ occur with asymptotic frequency $1/2$. This suggests a strong form of independence between additive and multiplicative structures in the Liouville function.