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 $d_\theta(n)$ gives $PD(\theta)$. We ask what remains of this picture when the integer is also rewarded according to its largest prime factor $P^+(n)$. For fixed $\theta,\beta>0$ and $\gamma\ge0$, we sample $N_x=n\le x$ with probability proportional to $d_\theta(n)\exp\{\beta H_\gamma(n)\}$, where $H_\gamma(n)=(\log P^+(n))^\gamma$ for $\gamma>0$, while $H_0(1)=0$ and $H_0(n)=1$ for $n\ge2$. Two phase transitions occur. At $\gamma=0$ the $PD(\theta)$ partition survives, whereas every fixed $\gamma>0$ forces one prime to carry asymptotically all logarithmic mass. The second transition, at $\gamma=1$, concerns the cofactor $R_x=N_x/P^+(N_x)$. With $a_x=\beta\gamma(\log x)^{\gamma-1}$, its law is asymptotic in total variation to $Q_{a_x}(m)=d_\theta(m)m^{-1-a_x}/\zeta(1+a_x)^\theta$. Thus, for $0<\gamma<1$, $a_x\log R_x$ has a $Gamma(\theta,1)$ limit, with shape $\theta$ and rate one, and the normalized logarithmic prime factors of $R_x$ have an independent $PD(\theta)$ limit; at $\gamma=1$, $R_x$ has a nondegenerate discrete limit; and for $\gamma>1$, $R_x=1$ with high probability. We also determine the joint limits involving $N_x/x$ and the normalizing constants, which include the classical Alladi--Erd\H{o}s asymptotic as a special case.