Divisor function inequality - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T10:13:52Z http://mathoverflow.net/feeds/question/65808 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/65808/divisor-function-inequality Divisor function inequality Alex Botros 2011-05-24T02:34:18Z 2011-05-24T04:34:48Z <p>I have been reading a paper on the Goldbach conjecture found at <a href="http://people.exeter.ac.uk/pt224/Goldbach.pdf" rel="nofollow">http://people.exeter.ac.uk/pt224/Goldbach.pdf</a>. 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 <a href="http://www.m-hikari.com/ijcms-2010/1-4-2010/mollinIJCMS1-4-2010.pdf" rel="nofollow">http://www.m-hikari.com/ijcms-2010/1-4-2010/mollinIJCMS1-4-2010.pdf</a> 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 <a href="http://people.exeter.ac.uk/pt224/Goldbach.pdf" rel="nofollow">http://people.exeter.ac.uk/pt224/Goldbach.pdf</a>. how can both be true? </p>