1

This question is asked based on curiosity. I came across few papers which find extreme values of RZF: $|\zeta(s)|$ on the critical line $Re(s)=\frac{1}{2}$. Though it is quite unclear whether it has got some scope.

When we are already striving to know whether the zeros lie elsewhere in the critical strip, few top mathematicians are trying to unravel the properties only on the critical line. Is it possible that the values or the growth of the function on the critical line definitely could lead us somewhere more exciting?

Thank you.

user511110
  • 312
  • $\zeta(1/2+it)$ is complex valued, do you mean $\zeta(1/2+it) \pi^{-it/2}\Gamma(1/4+it/2)$ which is real valued ? – reuns Oct 14 '20 at 16:40
  • I made the edits. Please check if that's clear. – user511110 Oct 14 '20 at 16:47
  • Do you mean perhaps Littlewood ? Apparently, the math community does not take his doubts whether RH is true serious. I am not deep enough in the matter to understand , why the community is so convinced that RH cannot be false . Either Littlewood is missing something , or the community denies important doubts . – Peter Oct 14 '20 at 17:04
  • The math community doesn't apparently just believe, but even tries to prove it. And over the course of proving something, they do find different amusing results. What's about Littlewood you are trying to say? – user511110 Oct 14 '20 at 17:14
  • 1
    First, we already know that the non-trivial zeros of the RZF are all on the critical strip. The RH is that they in fact lie on the critical line. Someone's "doubts" are not enough to say that it is false. Someone else's "beliefs" are not enough to say that it is true. Using computers, we have investigated far beyond anything dreamable in the days of Littlewood without ever finding a zero off the critical line, but that doesn't prove it is always true. To prove or disprove something one studies everything related until the matter is settles. Thus some study on the line, others study off. – Paul Sinclair Oct 14 '20 at 23:23
  • the question is unclear - it is well known that $|\zeta(1/2+it)|$ can be quite big infinitely many times (for example there is always a value as big as $e^{\frac{3\sqrt {\log T}}{4 \sqrt {\log \log T}}}$ in the interval $[T,2T]$) while Lindelof's conjecture mean that those values are conjectured not to be polynomially big; similarly, while it was proven that for any $1/2 < \sigma \le 1$, the values $\zeta(\sigma+it)$ are dense in the plane, the result is only conjectured for the critical line – Conrad Oct 15 '20 at 14:30

0 Answers0