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Shifted-prime divisors

Steve Fan, Carl Pomerance

Abstract

Let $ω^*(n)$ denote the number of divisors of $n$ that are shifted primes, that is, the number of divisors of $n$ of the form $p-1$, with $p$ prime. Studied by Prachar in an influential paper from 70 years ago, the higher moments of $ω^*(n)$ are still somewhat a mystery. This paper addresses these higher moments and considers other related problems.

Shifted-prime divisors

Abstract

Let denote the number of divisors of that are shifted primes, that is, the number of divisors of of the form , with prime. Studied by Prachar in an influential paper from 70 years ago, the higher moments of are still somewhat a mystery. This paper addresses these higher moments and considers other related problems.
Paper Structure (8 sections, 12 theorems, 115 equations, 3 tables)

This paper contains 8 sections, 12 theorems, 115 equations, 3 tables.

Table of Contents

  1. Introduction
  2. The constant $C$ in (\ref{['eq:SC']})
  3. The level sets of $\omega^*(n)$
  4. A lower bound for $\delta(\langle n\rangle)$
  5. Higher moments of $\omega^*(n)$
  6. A lower bound for the third moment
  7. The lower bound for the second moment
  8. A tail estimate

Key Result

Theorem 1

There exists a suitable constant $c>0$ such that for all $x\ge1$ and all sufficiently large $y$.

Theorems & Definitions (20)

  • Theorem 1
  • proof
  • Theorem 2
  • Lemma 1
  • proof
  • proof : Proof of Theorem \ref{['th:delta_k']}
  • Lemma 2
  • Theorem 3
  • proof
  • Lemma 3
  • ...and 10 more