About the YouTube video "The Riemann Hypothesis, Explained" About the video The Riemann Hypothesis, Explained at 658 secs and on:

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*What are the meanings of the curves (and their analytic forms) that create the spiral or heart shapes and that keep hitting the zeta zeros several times?


*What do these mean to have the curves keep hitting the zeta zeros multiple say N times?
What is the meaning of this N? if hitting N times? (or can the N be bounded finite or be infinite?)



 A: I recognize the spiral as being the curve parametrized by $\zeta(\frac{1}{2}+it)$ for $t$ in some finite interval $[0,T]$; the spiral is growing larger by making $T$ larger. The value $\zeta(\frac{1}{2})=1.46035...$ is the leftmost point on the spiral in the video, where it starts. If you look closely, the "critical line" $\mathrm{Re}(s)=\frac{1}{2}$ (the vertical orange line halfway between $0$ and $1$) is the one that turns into the spiral.
In general, the straight lines of the grid moments before are a "before" picture, and the collection of curves moments later are the "after" picture. That is, the curves are the images of $\zeta$ applied to the original straight lines. The dark teal curves are those that can be obtained using the original series definition of $\zeta$, while the orange ones are those that can only be obtained by using analytic continuation to extend the domain of $\zeta$.
(I am unsure but very curious how they smoothly transitioned the lines into curves.)
Every time the spiral hits the origin is a zero of the zeta function. So the multiplicity with which the spiral intersects the origin is the number of zeros $\zeta$ has on the critical line with imaginary part restricted to the interval $[0,T]$. As $T$ grows larger, this number will keep growing, forever.
