I have been searching for a bound of the divisor function $d(n)$, meaning the number of divisors of n. So far I have found that it can be bounded by $ d(n) \le e^{O(\frac{\log n}{\log \log n})}$ Wigert has proven the constant is $\log 2$ so $ d(n) \le e^{(\log 2+ o(1)) \frac{\log n}{\log \log n}} $ However, when I tried to check that bound on a computer, it did not seem right. I have drawn $d(n)$ (in blue), $e^{\frac{\log n}{\log \log n}}$ (in red) and $e^{\log 2 \frac{\log n}{\log \log n}}$ (in green) on the following graph:
Furthermore, when I plot $\frac{\log d(n)}{\log n / \log \log n}$, it does not seem to have $\log 2$ as a limit either. See this other graph:
It appears the constant 1 is a much better fit !
So my question is: is Wigert's bound only true for large $n$ (i.e larger than $10^6$) ?
My confusion comes from the second equation, which is clearly a bound (see this post on Tao's blog) and the result of Wikipedia, which is a limit.