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

TL;DR

This paper proves that the bias towards semiprimes with factors congruent to 3 modulo 4 exists from the very beginning, confirming a conjecture and combining explicit analytical methods with computational verification.

Contribution

It provides a complete proof of the conjecture on the modulo 4 bias for all semiprimes, using explicit estimates and computational checks for smaller cases.

Findings

Confirmed the bias for all x ≥ 9

Extended previous results with explicit bounds

Validated the conjecture through computational verification

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.