On the zeros of $\zeta(0.5 + it)$ for $t$ real

My question is a follow up to this question.

I am curious why in the answer to the linked question, we have to assume RH to be true. Aren't all the zeros of $$\zeta(0.5 + it)$$ for $$t$$ real on the critical line anyways? So can we then assume that $$\frac{Z'(t)}{Z(t)}$$ for $$t$$ real is monotone between consecutive zeros of $$Z$$ without invoking RH?

• If the Riemann hypothesis is false , the given expression has a non-real root. If it is true, the given expression has only real roots as desired. Jun 23 at 10:27
• Not following your logic. Of course $\zeta(0.5+it)$ has non real roots. No questions on that.
– sku
2 days ago
• "Aren't all the zeros of $\zeta(0.5+it)$ on the critical line anyways?" Well, if you're considering $t\mapsto \zeta(0.5+it)$, then the RH says that non-trivial zeros of this function lie on the real axis (i.e $0.5+it$ lies on the critical line $\text{Re}(s)=\frac{1}{2}$). So, if you're saying that all the zeros are on the critical line (I assume you mean on the real axis), then it means you're claiming the RH is true. 2 days ago
• For $t$ real, aren't all the zeros of $\zeta(0.5 + it)$ on the critical line? They have to be...Note I am not making any claims on $\zeta(\sigma + it)$ where $\sigma \neq 0.5$. I think the transformation you refer to is giving the possibility of $t$ being non-real. I am only interested in $t$ real.
– sku
2 days ago
• You seem to think that "for $t$ real, all zeros of $\zeta(0.5+it)$ are on the critical line" implies "$Z'(t)/Z(t)$ is monotone decreasing between the zeros of $Z(t)$". Can you justify that implication without RH? 2 days ago