On the distribution of products of two primes

• Published in: Journal of number theory Vol. 214; pp. 100 - 136
• Main Authors: Saad Eddin, Sumaia, Suzuki, Yuta
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.09.2020
• Subjects: Distribution of prime numbers; RSA integers; 11N25; 11N69; RSA integers; Distribution of prime numbers
• ISSN: 0022-314X; 1096-1658
• DOI: 10.1016/j.jnt.2020.04.018

For a real parameter r, the RSA integers are integers which can be written as the product of two primes pq with p<q≤rp, which are named after the importance of products of two primes in the RSA-cryptography. Several authors obtained the asymptotic formulas of the number of RSA integers. However, the previous results on the number of RSA integers were valid only in rather restricted ranges of the parameter r. Dummit, Granville and Kisilevsky found some bias in the distribution of products of two primes with congruence conditions. Moree and the first author studied some similar bias in the RSA integers, but they proved that at least for fixed r, there is no such bias. In this paper, we provide an asymptotic formula for the number of RSA integers available in wider ranges of r, and give some observations of the bias of such integers, by interpolating the results of Dummit, Granville and Kisilevsky and of Moree and the first author.