Stack Exchange network consists of 180 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.
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?
$\begingroup$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.$\endgroup$
$\begingroup$"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.$\endgroup$
$\begingroup$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.$\endgroup$
$\begingroup$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?$\endgroup$
By clicking βAccept all cookiesβ, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.
