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Difference between number of factors of different types.

Source: 2014 China TST 2 Day 2 Q6

March 20, 2014
number theory proposednumber theoryposet

Problem Statement

Let kk be a fixed even positive integer, NN is the product of kk distinct primes p1,...,pkp_1,...,p_k, a,ba,b are two positive integers, a,b≤Na,b\leq N. Denote S1={d∣S_1=\{d| d∣N,a≤d≤b,dd|N, a\leq d\leq b, d has even number of prime factors}\}, S2={d∣S_2=\{d| d∣N,a≤d≤b,dd|N, a\leq d\leq b, d has odd number of prime factors}\}, Prove: ∣S1∣−∣S2∣≤Ckk2|S_1|-|S_2|\leq C^{\frac{k}{2}}_k
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