On Shusterman's Goldbach-type problem for sign patterns of the Liouville function

Abstract

Let $\lambda$ be the Liouville function. Assuming the Generalised Riemann Hypothesis for Dirichlet $L$-functions (GRH), we show that for every sufficiently large even integer $N$ there are $a,b \geq 1$ such that $$ a+b = N \text{ and } \lambda(a) = \lambda(b) = -1. $$ This conditionally answers an analogue of the binary Goldbach problem for the Liouville function, posed by Shusterman. The latter is a consequence of a quantitative lower bound on the frequency of sign patterns attained by $(\lambda(n),\lambda(N-n))$, for sufficiently large primes $N$. We show, assuming GRH, that there is a constant $C > 0$ such that for each pattern $(\eta_1,\eta_2) \in \{-1,+1\}^2$ and each prime $N \geq N_0$, $$ |\{n < N : (\lambda(n),\lambda(N-n)) = (\eta_1,\eta_2)\}| \gg N e^{-C(\log \log N)^{6}}. $$ The proof makes essential use of the Pierce expansion of rational numbers $n/N$, which may be of interest in other binary problems.