Moving prime environments and arithmetic cluster limits
We study shrinking-target occurrences of completely multiplicative functions with arbitrarily moving phases on a fixed finite set of exceptional primes and independent Haar values at the remaining primes. After exact removal of the environment-only points, effective integer-subgroup profiles classify all subsequential marked cluster limits, while target-scale pair overlaps characterize Poisson convergence. The quantitative input is a direct Haar comparison with the moving germ kernel, uniform in the exceptional phases and with stretched-exponential error. For deterministic profiles and…
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