Sparse prime surgery in completely multiplicative sequences: Normality, rare-event realization and optimal repair
We study how changing prime signs affects fixed-word frequencies, rare-event laws and extremes of completely multiplicative sequences. The cost is $d_\sigma(f,g)=\sum_{f(p)\ne g(p)}p^{-\sigma}$. A known theorem of Frantzikinakis-Lemańczyk-de la Rue preserves normality at finite $d_1$; we give its elementary specialization. Using explicit arithmetic inputs from Long runs, we then establish three families of results. For each fixed $0<\sigma\le1$ and almost every base, reversible surgery realizes every stationary finite-intensity marked point-process law along selected scales, with signs,…
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