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

TL;DR

Under the assumption of GRH, the paper proves that for large even integers, there exist partitions with specific Liouville function sign patterns, providing a conditional analogue to Goldbach's problem for the Liouville function.

Contribution

It establishes a conditional lower bound on the frequency of sign patterns of the Liouville function for large primes, using Pierce expansion techniques.

Findings

Existence of partitions with specified Liouville sign patterns for large even integers.

Quantitative lower bounds on the occurrence of sign patterns for the Liouville function.

Application of Pierce expansion in analyzing binary sign pattern problems.

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.