Big biases amongst products of two primes (Q2810737)

The authors consider the asymptotic behavior of the number of odd integers that are the product of two primes, in the arithmetic progressions \(1\pmod{4}\) and \(3\pmod{4}\). The following assertion is the main result in the paper.NEWLINENEWLINELet \(\chi\) be a quadratic character of conductor \(d\). For \(\eta=-1\) or for \(\eta=1\) it holds that NEWLINE\[NEWLINE \frac{4\,\#\{pq\leqslant x:\chi(p)=\chi(q)=\eta\}}{\#\{pq\leqslant x:(pq,d)=1\}}=1+\frac{\eta}{\log\log x}\bigg(\sum\limits_{p}\frac{\chi(p)}{p}+o(1)\bigg). NEWLINE\]NEWLINE Various generalizations of this result are also presented.