Let $f(n)$ and $g(x,n)$ be real functions. If $$A=\sum_{n=1}^{\infty}f(n)\cos(g(x,n))\, \text{ and } \,B=\sum_{n=1}^{\infty}f(n)\cos\left(g(x,n)+\frac{\pi}{2}\right),$$ under what conditions does it occur that $A=B=0$?
I ask this question because I am looking at the sums $$\sum_{n=1}^{\infty}\frac{(\text{-}1)^{n+1}\cos(x\ln(n))}{n^a}\,\text{ and }\,\sum_{n=1}^{\infty}\frac{(\text{-}1)^{n+1}\cos(x\ln(n)+\frac{\pi}{2})}{n^a},$$ which are the real and imaginary parts of the Dirichlet eta function with complex argument $a+xi$ (with $0<a\leq\frac{1}{2}$). It seems from anecdotal evidence that these two sums only both equal zero when $a$ is $\frac{1}{2}$ (and $x$ is the imaginary part of any non-trivial zero of the zeta function).