Infinite sum of cos(ln(n))/n

Question

What would be the value of the infinite sum $$\sum_{n=1}^\infty\frac{\cos\ln n}{n}$$

Answer 1

For $ \Re(s) > 0$ $$\Re(\sum_{n=1}^N n^{-s}-\frac{1-(N+1)^{1-s}}{s-1}) = \Re(\sum_{n=1}^N \int_n^{n+1}\int_n^x st^{-s-1}dtdx)$$ The RHS gives the series $$\lim_{N \to \infty} \Re(\sum_{n=1}^N \int_n^{n+1}\int_n^x st^{-s-1}dtdx) \le \int_1^\infty |sx^{-s-1}|dx=\frac{|s|}{\Re(s)}$$ which converges absolutely and $$\lim_{N \to \infty}\Re(\sum_{n=1}^N n^{-s}-\frac{1-(N+1)^{1-s}}{s-1})$$ Converges.

Your series is obtained with $s=1+i$ so that $\Re(\frac{1-(N+1)^{1-s}}{s-1})$ is bounded but it oscillates without converging.