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I just came with the idea: what is the center of mass of the Riemann Zeta Function across the critical line? I mean: when you plot the parametric graph across the critical line, you get the famous plot:

Parametric Critical Line

And... Where would be placed the center of mass of that curve? Does it has any meaning?

Many thanks in advance!


EDIT (20/09/2018) I've done some computational work by averaging the coordinates of the graph (as far as I understand, this procedure can get you the center of mass). The following sequence of images is the plot of the pairs $(z, \zeta(z))$ when $z = 0.5+xi$ and $x \in [0, 100]$ (light grey). And starting at $x = 0$ and increasing by steps of 0.5, I've been computing the center of mass from $x=0$ up to $x = x_n$, which is shown as a magenta line:

enter image description here

It it easy to see that the sequentially computed center of mass seems to converge to zero in a fancy manner, which is shown here in these magnified versions of the same plot:

enter image description here enter image description here

So, even when a numerical computation is evidence of nothing, it seems that very first guess about the center of mass is that it exists and it converges to $0$. So, here it comes my question:

Does it exist the center of mass of $\zeta(0.5 +xi)$? It seems to converge to $0$: is it something already known? If so, is it somehow relevant? ...or am I missing something here?

Many thanks in advance!!

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    $\begingroup$ Wild guess: the mass center diverges to infinity $\endgroup$ – Wojowu Sep 18 '18 at 10:41
  • $\begingroup$ @Wojowu if you average all the vectors, you are right, but please, excuse my lack of precise language: there is some interpretation of center of mass so my question has a convergent response. I mean, if this shape would be a physical object, then you could find its center of mass. Does my question make any sense after this disclaimer? $\endgroup$ – Carlos Toscano-Ochoa Sep 18 '18 at 10:48
  • $\begingroup$ Keep in mind this "physical object" would still be infinite in size - claiming that such a thing has a well-defined center of mass is by far not obvious. What if instead of this plot you had just a half-line? What would its center of mass be? $\endgroup$ – Wojowu Sep 18 '18 at 10:51
  • $\begingroup$ @Wojowu That precise example you've mentioned has a divergent center of mass (so it doesn't have it), but this is because the coordinates of the points diverge. In the upper case, if that curve diverges at some point, then there would be a finite number of zeros, which is not the case... The center of mass should converge, and I'd like to know its coordinates. Could you please tell me the mathematical procedure to get the "center of mass" of a curve in the plane that is not dependent of the position of the curve, please? $\endgroup$ – Carlos Toscano-Ochoa Sep 18 '18 at 10:59
  • $\begingroup$ The plot of Riemann zeta function won't diverge like a line, yes, but it will still go arbitrarily far. Perhaps a more accurate picture is like this: imagine a curve which starts at (0,0), then goes to the point (1,0) and back, then (2,0) and back and so on. Just like a line it has no well-defined center of mass, yet it returns to zero infinitely often. $\endgroup$ – Wojowu Sep 18 '18 at 11:04

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