Tilting Billingsley's model toward a giant prime: two phase transitions

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 $\gamma \ge 0$, 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 $n\ge 2$. 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 $R_x=N_x\mathrm{/}{P}^{+}(N_x)$. With $a_x=\beta \gamma (\log x{)}^{\gamma -1}$, its law is asymptotic in total variation to $Q_{a_x}(m)=d_{\theta }(m)m^{-1-a_x}\mathrm{/}\zeta (1+a_x{)}^{\theta }$. 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 $N_x\mathrm{/}x$ and the normalizing constants, which include the classical Alladi–Erdős asymptotic as a special case.