Products of two proportional primes

TL;DR

This paper studies RSA-integers formed by proportional primes, providing a precise asymptotic count and extending previous results on prime distributions in such products.

Contribution

It establishes a refined asymptotic formula for RSA-integers, improving upon earlier estimates and extending prime distribution results to this specific class of integers.

Findings

Derived a more accurate asymptotic count of RSA-integers up to x.

Extended prime distribution results to RSA-integers with proportional primes.

Provided a strong form of the prime number theorem for this context.

Abstract

In RSA cryptography numbers of the form $pq$, with $p$ and $q$ two distinct proportional primes play an important role. For a fixed real number $r>1$ we formalize this by saying that an integer $pq$ is an RSA-integer if $p$ and $q$ are primes satisfying $p<q\le rp$. Recently Dummit, Granville and Kisilevsky showed that substantially more than a quarter of the odd integers of the form $pq$ up to $x$, with $p, q$ both prime, satisfy $p\equiv q\equiv 3\pmod{4}$. In this paper we investigate this phenomenon for RSA-integers. We establish an analogue of a strong form of the prime number theorem with the logarithmic integral replaced by a variant. From this we derive an asymptotic formula for the number of RSA-integers $\le x$ which is much more precise than an earlier one derived by Decker and Moree in 2008.