Problem

Source: 2014 China TST 2 Day 2 Q6

Tags: number theory proposed, number theory, poset



Let $k$ be a fixed even positive integer, $N$ is the product of $k$ distinct primes $p_1,...,p_k$, $a,b$ are two positive integers, $a,b\leq N$. Denote $S_1=\{d|$ $d|N, a\leq d\leq b, d$ has even number of prime factors$\}$, $S_2=\{d|$ $d|N, a\leq d\leq b, d$ has odd number of prime factors$\}$, Prove: $|S_1|-|S_2|\leq C^{\frac{k}{2}}_k$