Poisson Approximation of Prime Divisors of Shifted Primes.

The article develops a Kubilius-type probabilistic model for the prime factorization of shifted primes, numbers of the form \( p + a \) where \( p \) is prime and \( a \neq 0 \) is fixed. It establishes a total variation distance bound between the distribution of prime factors of shifted primes and an explicit independent random variable model, showing that prime factors in disjoint sets behave like independent Poisson variables under certain hypotheses on the distribution of primes in arithmetic progressions (Hypothesis \( Z(\gamma) \)). This leads to a transference principle asserting that many statistical properties of prime factors of random integers also hold for shifted primes, excluding those strongly dependent on the smallest or largest prime factors. The work includes applications such as deriving the law of the iterated logarithm for the count of prime factors of shifted primes from the analogous result for integers.