Chebyshev Bias for Largest Prime Factors

Abstract

Let $P^+(n)$ be the largest prime factor of $n$, and let $\chi=\chi_{-4}$ be $1$ on primes $1\pmod4$ and $-1$ on primes $3\pmod4$. For fixed $k\ge2$ we study \[ D_k(x)=\sum_{\substack{n\le x\\ \Omega(n)=k}}\chi(P^+(n)). \] Thus $D_k(x)$ compares the two residue classes according to the largest prime factor of integers having exactly $k$ prime factors, counted with multiplicity. Assuming RH for $\zeta(s)$ and $L(s,\chi)=\beta(s)$, we prove a pointwise explicit formula. The main term is a fixed negative contribution plus an absolutely convergent oscillating sum over the zeros $\rho=\frac12+i\gamma$ of $L(s,\chi)$. The ordering of the prime factors produces $k$ Perron denominators, and the principal coefficient of a zero is $O_k((1+|\gamma|)^{-k})$. We then show that the total size of all zero terms is strictly smaller than the fixed contribution. Hence $D_k(x)<0$ for all sufficiently large $x$. The analogous problem without fixing $k$ is still open.