Chowla’s conjecture for multidimensional Liouville correlations

Prove that for the Liouville function \(\lambda\) and every non-empty finite set \(H=\{h_1,\ldots,h_d\}\) of non-negative integers, the multidimensional correlation average satisfies \(\lim_{x\to\infty}x^{-1}\sum_{n\leq x}\lambda(n+h_1)\cdots\lambda(n+h_d)=0\).

Background

The paper studies correlations of completely multiplicative functions taking values in $\{\pm1\}$, with the Liouville function corresponding to the choice in which every prime is assigned the value $-1$. Chowla’s conjecture predicts that the values of the Liouville function at any fixed collection of distinct shifts behave sufficiently independently that their normalized correlation tends to zero.

The paper notes that the conjecture is known in full only for $d=1$, where it is equivalent to the prime number theorem. The multidimensional cases remain unresolved and motivate the paper’s analysis of related functions $\lambda_P$ and restricted classes of prime sets.