Long runs and rare patterns of a random completely multiplicative function
Let $f$ be a random completely multiplicative function with independent symmetric signs at the primes. We study rare constant runs and prescribed words at logarithmic lengths, despite long-range multiplicative identities. A uniform weighted two-window square-relation estimate, with critical bound $O_\varepsilon(N^{5/3+\varepsilon})$, separates exact rational relations from residual components. At critical intensity, this estimate gives Poisson comparisons retaining positions, exact excess lengths and signs. For dictionaries, uniform bounds under size and overlap conditions are complemented…
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