Let $d(m)$ denote the number of divisors of $m$ and let $N$ be a large integer. Then we have $$\sum_{n \leq N}\frac{d(n)}{n} \geq \left(\sum_{n \leq \sqrt{N}}\frac{1}{n}\right)^{2} \sim \log^{2}N.$$ What prevents me from doing $$\sum_{n \leq N}\frac{d(n)}{n} \geq \left(\sum_{n \leq N^{1/k}}\frac{1}{n}\right)^{k} \sim \log^{k}N$$ for every integer $k$?
Question about number theory asymptotic proof
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number-theory