Reflected sign patterns of the Liouville function at every large total
For signs $s,t$, let $N_{s,t}(N)$ count the ordered representations $N=n+(N-n)$ with $\lambda(n)=s$ and $\lambda(N-n)=t$. We prove that, for every fixed $\varepsilon>0$, all four counts are at least $c_\varepsilon N^{1-\varepsilon}$ at every sufficiently large integer total. The constants are ineffective through Siegel's theorem. The same conclusion holds uniformly for completely multiplicative sign functions whose positive primes satisfy $\sum_{f(p)=1}p^{-\theta}\le M$, with fixed $0<\theta<1$ and $M$. Separately, at each sufficiently large prime $p$, both patterns with product $(-1/p)$…
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