I have been reading a paper on the Goldbach conjecture found at http://people.exeter.ac.uk/pt224/Goldbach.pdf. At one point, the author (Paul Truman), states: Let $z=N^{1/8}$, then $$\sum_{w\leq z}\frac{d(w)}{w}\gg(\log(z))^2\gg(\log N)^2$$ where $d(w)$ counts all the positive divisors of $w$. I am assuming that there's a mistake in the second part of the inequality $(\log(z))^2\gg(\log N)^2$, but this is not the first time I've encountered such a claim: at http://www.mhikari.com/ijcms2010/142010/mollinIJCMS142010.pdf the author claims (in his proof of the upper bound on the twin prime counting function) that $$\sum_{\substack{d\leq N^{1/3}\\ d \\ odd}}\frac{f(d)}{d} \geq (\log N)^2$$ where $f(2)=1, f(p)=2$ for all odd primes $p$. This inequality I have seen proved (though I can't recall where) by saying that $f(w)/w\geq d(w)/w$. A lot of people are saying that $$\sum_{w\leq z}\frac{d(w)}{w}\gg(\log(z))^2$$ but others are saying that $$\sum_{w\leq z}\frac{d(w)}{w}=\frac{\log^2(z)}{2}+O(\log x)$$ including http://people.exeter.ac.uk/pt224/Goldbach.pdf. how can both be true?
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