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 $a,b\ge 1$ 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 $N\ge N_0$, $$|\{n<N:(\lambda (n),\lambda (N-n))=({\eta }_1,{\eta }_2)\}|\gg N{e}^{-C(\log \log N{)}^6}.$$ The proof makes essential use of the Pierce expansion of rational numbers $n/N$, which may be of interest in other binary problems.

23 pages; comments welcome!