Integers divisible by a large shifted prime

TL;DR

This paper determines the growth rate of integers up to x that are divisible by shifted primes p-1 with p greater than y, refining previous bounds and providing exact asymptotics for all relevant ranges.

Contribution

It establishes the precise order of growth of N(x,y) for all x ≥ 2y ≥ 4, improving upon recent bounds by McNew, Pollack, and Pomerance.

Findings

Derived the exact asymptotic growth of N(x,y) for all x ≥ 2y ≥ 4.

Improved previous bounds on the distribution of integers divisible by shifted primes.

Extended the understanding of the density of integers divisible by primes minus one.

Abstract

Let $N(x,y)$ denote the number of integers $n\le x$ which are divisible by a shifted prime $p-1$ with $p>y$, $p$ prime. Improving upon recent bounds of McNew, Pollack and Pomerance, we establish the exact order of growth of $N(x,y)$ for all $x\ge 2y\ge 4$.