Tilting Billingsley's model toward a giant prime: two phase transitions
Abstract
Billingsley's theorem states that the normalized logarithms of the prime factors of a uniform random integer in [1,x] converge to the Poisson--Dirichlet law PD(1), while weighting an integer n by the generalized divisor function dθ(n) gives PD(θ). We ask what remains of this picture when the integer is also rewarded according to its largest prime factor P+(n). For fixed θ,β>0 and γ≥0, we sample Nx=n≤ x with probability proportional to dθ(n)exp{βHγ(n)}, where Hγ(n)=(log P+(n))γ for γ>0, while H0(1)=0 and H0(n)=1 for n≥2. Two phase transitions occur. At γ=0 the PD(θ) partition survives, whereas every fixed γ>0 forces one prime to carry asymptotically all logarithmic mass. The second transition, at γ=1, concerns the cofactor Rx=Nx/P+(Nx). With ax=βγ(log x)γ−1, its law is asymptotic in total variation to Qax(m)=dθ(m)m−1−ax/ζ(1+ax)θ. Thus, for 0<γ<1, axlog Rx has a $Gamma(θ,1)$ limit, with shape θ and rate one, and the normalized logarithmic prime factors of Rx have an independent PD(θ) limit; at γ=1, Rx has a nondegenerate discrete limit; and for γ>1, Rx=1 with high probability. We also determine the joint limits involving Nx/x and the normalizing constants, which include the classical Alladi--Erdős asymptotic as a special case.
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BibTeX
@misc{sun2026tilting,
title = {{Tilting Billingsley's model toward a giant prime: two phase transitions}},
author = {Wen Sun},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11735},
eprint = {2609.11735},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11735},
}
RIS
TY - UNPB
TI - Tilting Billingsley's model toward a giant prime: two phase transitions
AU - Wen Sun
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11735
UR - https://arxiv.org/abs/2609.11735
AB - Billingsley's theorem states that the normalized logarithms of the prime factors of a uniform random integer in [1,x] converge to the Poisson--Dirichlet law PD(1), while weighting an integer n by the generalized divisor function d_θ(n) gives PD(θ). We ask what remains of this picture when the integer is also rewarded according to its largest prime factor P^+(n). For fixed θ,β>0 and γ≥0, we sample N_x=n≤ x with probability proportional to d_θ(n)exp{βH_γ(n)}, where H_γ(n)=(log P^+(n))^γ for γ>0, while H_0(1)=0 and H_0(n)=1 for n≥2. Two phase transitions occur. At γ=0 the PD(θ) partition survives, whereas every fixed γ>0 forces one prime to carry asymptotically all logarithmic mass. The second transition, at γ=1, concerns the cofactor R_x=N_x/P^+(N_x). With a_x=βγ(log x)^γ−1, its law is asymptotic in total variation to Q_a_x(m)=d_θ(m)m^−1−a_x/ζ(1+a_x)^θ. Thus, for 0<γ<1, a_xlog R_x has a $Gamma(θ,1)$ limit, with shape θ and rate one, and the normalized logarithmic prime factors of R_x have an independent PD(θ) limit; at γ=1, R_x has a nondegenerate discrete limit; and for γ>1, R_x=1 with high probability. We also determine the joint limits involving N_x/x and the normalizing constants, which include the classical Alladi--Erdős asymptotic as a special case.
ER -