A simple counterexample to Havil's ``reformulation'' of the Riemann hypothesis (Q442466)

In his book [Gamma: Exploring Euler's constant. Princeton, NJ: Princeton University Press (2003; Zbl 1023.11001), pbk (2009; Zbl 1177.11001)] \textit{J. Havil} claims that the following conjecture is a ``tantalizing simple reformulation'' of the Riemann Hypothesis (RH): NEWLINENEWLINEHavil's conjecture: If NEWLINE\[NEWLINE\sum_{n=1}^\infty \frac{(-1)^n}{n^a}\cos(b\ln n)=0\quad \text{and}\quad \sum_{n=1}^\infty \frac{(-1)^n}{n^a}\sin(b\ln n)=0NEWLINE\]NEWLINE for some pair of real numbers \(a\) and \(b\), then \(a=1/2\). NEWLINENEWLINEIn this nicely written article the author first explains the RH and its connection with Havil's conjecture. Then he gives a counter-example to Havil's conjecture (but not to RH) by setting the pair of real numbers \(a=1\) and \(b=2\pi/\ln 2\).NEWLINENEWLINEIn the last section he gives a corrected version of Havil's conjecture changing the then part to ``\(a=1/2\) or \(a=1\)'' and proves that new conjecture is true if and only if the RH is true.