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The Riemann Zeta function on the critical line goes through a series of looping orbits, and we usually find one zero between two successive "apogees".

But is it possible to have a close approach to zero (a "perigee") that misses zero by a small error? Fig: Plot of zeta along critical line

Praveen B.
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  • I would bet that this is equivalent to the Riemann hypothesis (and that someone might likely be able to answer this on this site) - a pair of zeros off of the critical line would manifest itself as a "miss" (e.g. think about the function $(x+i)(x-i)$ evaluated on $\mathbb R$). – Milo Brandt Dec 28 '19 at 03:26

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Plot the curve in $\Bbb{R}^2$ given by $\zeta(1/2+it),t\in [A,B]$

As you see sometimes the curve crosses itself at zero, sometimes somewhere else, and sometimes there is a zero in each pair of crossing of the real axis, sometimes there are two consecutive crossings without zeros. To understand what is happening you need to look at the density of zeros theorems, in particular the error term $O(\log t)$ for the (continuous branch) of $arg\ \zeta(1/2+it)$ obtained from the functional equation showing that $\zeta(1/2+it)e^{-i\vartheta(t)}$ is real. The $e^{-i\vartheta(t)}$ term is well-understood, it has a simple asymptotic expansion, whereas the location of the zeros is quite erratic.

enter image description here

      t = [2200:0.04:2203.4]; L = length(t); z = zeros(1,L); for l=1:L, z(l) = zeta(1/2+i*t(l)); end; plot(z);

https://octave-online.net/

reuns
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  • Where can I learn more about the $e^{-i\vartheta(t)}$ ? I don't read French, so I couldn't get much from that Wiki link. Maybe this term has a name that I can Google, or if you could provide a link to a site in English? – Praveen B. Dec 29 '19 at 02:19
  • https://en.wikipedia.org/wiki/Z_function There is a chapter on the density of zeros in every book on $\zeta(s)$ – reuns Dec 29 '19 at 02:22