This property is rather elementary, and not at all specific to $\zeta$, so I am wondering if it has any value in studying the zeros of the Riemann zeta function in the critical strip. It is a well known result? I can provide a proof sketch if you are interested, and it has been checked numerically.

If $\zeta(s)=0$, with $s=\sigma +it$ and $0<\sigma<1$ then for all real $\theta$, we have

$$\sum_{n=1}^{\infty}(-1)^{n+1}\frac{\cos(\theta+t\log n)}{n^\sigma}=0.$$

This would be true even if by chance, one of the zeroes is outside the critical line $\sigma=\frac{1}{2}$. In particular let $t_0$ be the imaginary part of a zero of $\zeta(s)$. Then in order to find all $\sigma$'s such that $\zeta(\sigma +it_0)=0$, we only need to focus on the $\sigma$'s that satisfy the following equation for every $\theta$:

$$\sum_{n=1}^{\infty}(-1)^{n+1}\frac{\cos(\theta+t_0\log n)}{n^\sigma}=0.$$

Of course the Riemann Hypothesis (RH) implies that $\sigma$ must be equal to $\frac{1}{2}$, but my assertion is true even if RH is not true. So if you prove that my equality can only happen if $\sigma=\frac{1}{2}$, then you would have proved RH. I have several strong reasons to believe that my equality does not lead to a proof of RH, yet I am wondering if it has any value.