Poisson Approximation of prime divisors of shifted primes

TL;DR

This paper develops a probabilistic model for the distribution of prime factors of shifted primes, showing they behave like independent Poisson variables and establishing a connection to the prime factorization of random integers.

Contribution

It introduces a Poisson approximation framework for prime factors of shifted primes and proves a total variation distance estimate for the model's accuracy.

Findings

Prime factors of shifted primes in disjoint sets behave like independent Poisson variables.

A transference principle links the prime factorization of shifted primes to that of random integers.

The model accurately approximates the distribution of prime factors of shifted primes.

Abstract

We develop an analog for shifted primes of the Kubilius model of prime factors of integers. We prove a total variation distance estimate for the difference between the model and actual prime factors of shifted primes, and apply it to show that the prime factors of shifted primes in disjoint sets behave like independent Poisson variables. As a consequence, we establish a transference principle between the anatomy of random integers up to x and of random shifted primes p+a with p < x.