Long runs and rare patterns of a random completely multiplicative function
Abstract
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 Oε(N5/3+ε), 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 by typical bounds obtained by averaging the distance after fixing the dictionary. In a quantitatively controlled regime of diverging intensity, an exact one-word coupling by largest-odd-prime pivots compares the full signed run field under prescribed small-prime conditioning. For fixed admissible window parameters and almost every fixed realization of f, run-start counts in prescribed near-macroscopic windows satisfy an empirical Poisson law along dyadic scales. In the exceptionally rare regime, a relative marked comparison separates the boundary event from bulk occurrences. Conditional on existence, the first location, sign and overshoot have a two-source lattice law with scale-dependent weights. The two overshoots use different clocks: prime rank at the boundary and integer distance in the bulk.
Related
- Low moments of automorphic random multiplicative function sums
- Pythagorean triples in level sets of completely multiplicative functions
- On Completely Positive and Semidefinite Programs for Moment Bounds on Maximum of Random Quadratics
- Almost sure upper bound for random multiplicative functions
- Estimation beyond missing (completely) at random
BibTeX
@misc{pouly2026long,
title = {{Long runs and rare patterns of a random completely multiplicative function}},
author = {Brice Pouly},
year = {2026},
month = sep,
doi = {10.33774/coe-2026-z3l74-v3},
url = {https://doi.org/10.33774/coe-2026-z3l74-v3},
note = {CC BY},
}
RIS
TY - UNPB
TI - Long runs and rare patterns of a random completely multiplicative function
AU - Brice Pouly
PY - 2026
DA - 2026/09/24
DO - 10.33774/coe-2026-z3l74-v3
UR - https://doi.org/10.33774/coe-2026-z3l74-v3
AB - 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 Oε(N5/3+ε), 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 by typical bounds obtained by averaging the distance after fixing the dictionary. In a quantitatively controlled regime of diverging intensity, an exact one-word coupling by largest-odd-prime pivots compares the full signed run field under prescribed small-prime conditioning. For fixed admissible window parameters and almost every fixed realization of f, run-start counts in prescribed near-macroscopic windows satisfy an empirical Poisson law along dyadic scales. In the exceptionally rare regime, a relative marked comparison separates the boundary event from bulk occurrences. Conditional on existence, the first location, sign and overshoot have a two-source lattice law with scale-dependent weights. The two overshoots use different clocks: prime rank at the boundary and integer distance in the bulk.
ER -