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 $1(\mathrm{mod}4)$ and $-1$ on primes $3(\mathrm{mod}4)$. For fixed $k\ge 2$ we study \[ D_k(x)=\sum_{\substack{n\le x\\ Ω(n)=k}}χ(P^+(n)). \] 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 $\rho =\frac{1}{2}+i\gamma$ 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.

15 pages