Proof of the Complete Presence of a Modulo 4 Bias for the Semiprimes

Abstract

In 2016, Dummit, Granville, and Kisilevsky showed that the proportion of semiprimes (products of two primes) not exceeding a given $x$, whose factors are congruent to $3$ modulo $4$, is more than a quarter when $x$ is sufficiently large. They have also conjectured that this holds from the very beginning, that is, for all $x \geq 9$. Here we give a proof of this conjecture. For $x\geq 1.1 \cdot 10^{13}$ we take an explicit approach based on their work. We rely on classical estimates for prime counting functions, as well as on very recent explicit improvements by Bennett, Martin, O'Bryant, and Rechnitzer, which have wide applications in essentially any setting involving estimations of sums over primes in arithmetic progressions. All $x < 1.1 \cdot 10^{13}$ are covered by a computed assisted verification.