Let $d(n)$ be number of positive divisor of natural number $n$. It is known from $d(n)=O(n^{\varepsilon})$ for every $\varepsilon>0$, we can deduce that $d(n)=n^{C/(\log \log n)}$ for some constant $C$. I am curious whether there is a proof for this fact directly and deduce $d(n)=O(n^{\varepsilon})$ as a corollary. Does anybody know how to do this or have reference to the proof?
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I put a summary at https://math.stackexchange.com/questions/3953080/what-are-some-upper-bounds-for-the-number-of-factors-of-a-number-a-proof-would/3953231#3953231 where the information is from Nicolas http://math.univ-lyon1.fr/~nicolas/publications.html and particularly pdf http://math.univ-lyon1.fr/~nicolas/hcnrevisited.pdf – Will Jagy Mar 16 '24 at 02:10
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@WillJagy thank you. – Laurence PW Mar 16 '24 at 13:14