On a Goldbach-Type Problem for the Liouville Function

Abstract

Abstract Let $\lambda $ denote the Liouville function. We show that for all $N \geq 11$, the (non-trivial) convolution sum bound $$ \begin{align*} & \left|\sum_{n < N} \lambda(n) \lambda(N-n)\right| < N-1 \end{align*} $$ holds. We also determine all $N$ for which no cancellation in the convolution sum occurs. This answers a question posed at the 2018 AIM workshop on Sarnak’s conjecture.