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

I would start with this:
$$0\leq \left|\frac{1}{\log(n)}\sum_{k=0}^n\frac{\cos(\sin(2\pi\log(k)))}{k}\right|\leq \frac{1}{\log{n}}\sum_{k=0}^n\frac{1}{k}$$
