A Lower Bound For Biases Amongst Products Of Two Primes

TL;DR

This paper proves a conjecture related to the maximum bias in prime number races involving products of two primes and quadratic characters, advancing understanding of prime distribution biases.

Contribution

It establishes a conjectured result on the maximum bias in P2 prime races, specifically for primes with quadratic character evaluations, filling a gap in number theory.

Findings

Proves the conjectured maximum bias in P2 races.

Provides new bounds for prime distribution biases.

Advances theoretical understanding of prime factorization biases.

Abstract

We establish a conjectured result of Dummit, Granville and Kisilevsky for the maximum bias in P2 races, which compare the number of P2's $\leq x$ whose prime factors, evaluated at a given quadratic character, take specifc values.