Upper Bound on Number of Factors

Question

Is there a theorem, lemma, or proof somewhere that proves an upper bound for the number of factors that a number can have?

If not, would it be fairly trivial to prove that it is $log_2 n$?

Answer 1

Let ${d(n)}$ denote the number of factors of $n$.

There is a useful bound which has applications in many area:

${d(n)\le n^{O(1/\log \log n)} = \exp(O(\frac{\log n}{\log\log n}))}$.

Maybe Tao's blog can help you understand the answer better.

https://terrytao.wordpress.com/2008/09/23/the-divisor-bound/