math.NT — Number Theory

Sparse prime surgery in completely multiplicative sequences: Normality, rare-event realization and optimal repair

Contributed by Brice Pouly

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, exact excess lengths and converging actual intensity, at arbitrarily small $d_\sigma$ cost and with a prescribed prefix unchanged. Almost surely over all integer scales, the minimum number of prime changes needed for a run of length $\lceil c\log_2N\rceil$ is asymptotic to $c\alpha_c\log_2N$ for $1<c<3/2$, where $\alpha_c\in(0,1/2)$ satisfies $H_2(\alpha_c)=1-1/c$ and $H_2(u)=-u\log_2u-(1-u)\log_2(1-u)$. At each scale, with probability tending to one, symbolic and private large-prime repair maxima coincide on the same random base for $k_N=o(\log N)$; unrestricted prime repair agrees when $k_N=o(\log\log N)$. This comparison transfers the classical lattice laws of runs with errors to adaptive prime repair. Elementary programming from arbitrary bases and an autonomous finite-range-dependent process construction complement these results.