On the distribution of products of two primes

For a real parameter $r$, the RSA integers are integers which can be written as the product of two primes $pq$ with $p<q\leq rp$, 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 the RSA integers. However, the previous results on the number of the RSA integers were valid only in a rather restricted range 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 the RSA integers available in wider ranges of $r$, and give some observations of the bias of the RSA integers, by interpolating the results of Dummit, Granville, and Kisilevsky and of Moree and the first author.