math.NT — Number Theory

Reflected sign patterns of the Liouville function at every large total

Contributed by Brice Pouly

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 $(-1/p)$ have at least $p/(4(\log p)^{96})$ representations. All four satisfy this stronger bound in six reduced classes modulo $24$ and outside one very sparse family of primes. The proof combines finite reflection-to-dilation propagation, a short-product spectral test, reconstruction from the units, and signed counting of prime disagreements. For the uniform extension, candidates are united before a common exceptional population is removed. None of the conclusions gives a positive-density bound or cancellation of the reflected correlation at every total.