Yes, all roots are simultaneous. Also, there is only one line per root. Your graph has 2 lines/root because it is in 3-D. In 4-D, as a complex functions are graphed in entirety, there is only 1 line/root. The trivial zeroes manifest this, as there can be multiple roots/line. (The trivial zeroes line flashes instantaneously at t=0). I appreciate your work as being the first detailed graph I've seen of the Riemann surfaces(3-D, yours 2 spatial, 1 time) in the critical strip available for anyone to see at home. Someone will become the first person to graph RZ in 4-D. Go for it.
@@zaphodbeeblebrox5511 Thanks for the response :) Then, @3:23 all lines will pass through (0,0) only when alpha is a negative even integer, not in other cases.
@@JwalinBhatt Actually no... The trivial zeros are on x-axis for b=0. Having a red line means that b increases. So, the red line only shows the zeros for b>0. In 3:23 you are correct that a
@@zaphodbeeblebrox5511 Thanks for the explanation, I'm trying to understand. You've done a excellent job exploring the real component, keeping imaginary fixed. You're right, I also hadn't found such a video before so it was intriguing to watch :) Let me phrase my previous comment differently. The squiggles @3:39 (a=-2.9) correspond to the spiral @1:23 (a=-2.9). The squiggles have beta along the x axis, so the left most point corresponds beta=0, so we have zeta(-2.9 + 0i) which is approximately 0.00879463 but not exactly 0. Hence, the centremost point of the corresponding spiral would end up at (0.00879, 0) but not exactly at the origin. And this would be the case for all values of alpha that aren't a negative integer.
The jitter of the white box is quite distracting. Maybe a more stable and smoother x/y axis scaling could help? 🤗 And a slow-down around the critical strip, as so much is happening here.
Your observations are spot on! I don't really know how to make videos and I also had some problems with mathematica. I was surprised to see that there was no video on YT with this animation and I thought I should give it a try. Thanks for the feedback
@@zaphodbeeblebrox5511 I tried to do this a few years ago with javascript but never really managed to nail it the way you did. I was so happy to see your animations, my heart jumped 😃
