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We construct a probabilistic model for the number of divisors of a random uniform integer that converges in the mod-Poisson sense to the same limiting function as its original counterpart, the one arising in the Sathé–Selberg theorem. This construction involves a conditioning and gives an alternative perspective to the usual paradigm of “hybrid product” models developed by Gonek, Hughes and Keating in the case of the Riemann Zeta function.

A Penalised Model Reproducing the Mod-Poisson Fluctuations in the Sathé–Selberg Theorem