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Poisson Approximation of prime divisors of shifted primes

Kevin Ford

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.

Poisson Approximation of prime divisors 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.
Paper Structure (10 sections, 15 theorems, 88 equations)

This paper contains 10 sections, 15 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Kubilius' model of prime factors of integers
  3. A Kubilius type model for shifted primes
  4. Poisson approximation of prime factors of shifted primes
  5. An application
  6. Total variation distance inequalities
  7. Sieving by small primes
  8. The proof of Theorem \ref{['thm:main']}
  9. Poisson approximation: the proof of Theorem \ref{['thm2']}
  10. The transference principle: proof of Theorem \ref{['thm:transference']}

Key Result

Theorem 1

Assume Hypothesis $Z(\gamma)$. Fix $a\ne 0$, $A>0$ and $0<\alpha < \gamma$. Then, for $2\leqslant y\leqslant x$ we have the implied constant in the $\ll-$symbol depending only on $a,A$ and $\alpha$.

Theorems & Definitions (25)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • Corollary 5
  • proof
  • Theorem 6
  • proof
  • Lemma 2.1
  • proof
  • ...and 15 more