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In analytic number theory, it is common to prove that $$\sum_{n\leq N} \frac{a_n}{n} = o(\log N)\tag{$\star$}\label{476699_star}$$ for some sequence $\{a_n\}_{n=1}^\infty$, $a_n\in \mathbb{C}$. (It is ...

Let $0<\alpha <1$ and $\beta > 0$. Consider the mapping $$F(\alpha, \beta) = \sum_{n=1}^{\infty}{\dfrac{(-1)^{n-1}\biggl( \cos \left(\beta\ln(n)\right)\biggr)}{n^{\alpha}}}.$$ Can we prove $F(...