$\lim_{n\to\infty} \frac{1}{\log(n)}\sum _{k=1}^n \frac{\cos (\sin (2 \pi \log (k)))}{k}$ What tools would you gladly recommend me for computing precisely the limit below? Maybe a starting point?
$$\lim_{n\to\infty} \frac{1}{\log(n)}\sum _{k=1}^n \frac{\cos (\sin (2 \pi  \log (k)))}{k}$$

Using the standard estimate for the harmonic numbers,
$$
\left| \frac{1}{\log n} \sum_{k=1}^n \frac{\cos(\sin(2\pi \log(k)))}{k}\right| \leq \frac{1}{\log n} \sum_{k=1}^n\frac{1}{k} = \frac{1}{\log n}(\log(n) + \gamma + O(1/n)).
$$
So in the limit,
$$
\left| \lim_{n\to \infty}\frac{1}{\log n} \sum_{k=1}^n \frac{\cos(\sin(2\pi \log(k)))}{k} \right| \leq 1.
$$
This is a pretty elementary bound, so you were probably already aware of it, but hopefully it helps.
