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Contemplating a question on math.SE, I have stumbled on this:

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Here, the point labeled $n$ is that root of the $n$th Bernoulli polynomial which has smallest positive imaginary part.

Does anyone know an explanation of this pattern? Specifically, some periodicity modulo 5 must be involved somehow.

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  • $\begingroup$ I wouldn't be surprised that connections with physics exist. $\endgroup$ Commented May 20, 2017 at 10:23

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It is known that $B_n(x)$ has $\asymp \frac2{\pi e}n$ real roots (see Veselov, A. & Ward, J. On the real roots of the Bernoulli polynomials and the Hurwitz zeta-function) Probably it means that the first "positive" complex root becomes real after each $\frac{\pi e}2=4.27\ldots$ steps.

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  • $\begingroup$ Very interesting observation, but probably at $n=300$ the fact that $\frac{\pi e}2$ is substantially smaller than $5$ should already show itself? I mean, all these curves contain arithmetic progressions with step $5$ all the way until 300... $\endgroup$ Commented May 20, 2017 at 11:08
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For the limiting curve formed by the complex zeros of the Bernoulli polynomials, see Goh, Boyer, On the Zero Attractor of the Euler Polynomials (Adv. Appl. Math. 2007, doi:10.1016/j.aam.2005.05.008). Actually, the authors treat the closely related Euler polynomials and say that Bernoulli polynomials "are easily handled with the techniques in this paper." The connection is made explicit on page 21 of http://www.math.drexel.edu/~rboyer/talks/MIT_FINAL.pdf ; the limiting curve for Bernoulli polynomials is half that of Euler polynomials. Perhaps the answer to your question can be obtained from this curve.

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(too long for a comment)

I think there is nothing happening modulo 5, at least not in a number-theoretical sense. It is first of all the visual impression because of the fact that most of the segments linking a point $P_n$ to $P_{n+5}$ have a slope close to 0 (about $-.06$), so they are geometrically closer than, say, $P_n$ and $P_{n+6}$, meaning that we easily perceive those patterns as "almost horizontal" lines (depending on the scaling), slightly descending.
IMO what is more striking is the fact that the real parts show a "very close to linear" progression of about 1/17 from $P_n$ to $P_{n+1}$, and that every 5 of them (from time to time 6), the imaginary part, usually increasing, "jumps" back to the bottom. The top points (11, 32, 53, ...) are 21 apart (from time to time 26), with the points 116, 184, 231..., which "should" be on top but cause a distance 26 instead of 21, jumping even lower. This reminds me a lot of what happens when projecting a sinusoidal curve wrapped around a cylinder (Lissajous curves): If you choose points on the sinus curve with linearly progressing x-coordinates, their projections will behave very similarly as the $P_n$'s here.
There should be some constant $\approx 5.2$ for the average period of the jumps, and so the main question would be how to find this constant, which does not seem to be related to $\frac{\pi e}2$.

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  • $\begingroup$ But do you see an explanation of the fact that, although jumps of the imaginary part occur sometimes after 5 and sometimes after 6 steps, the resulting strings invariably contain mod 5 arithmetic progressions? $\endgroup$ Commented May 26, 2017 at 16:22
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    $\begingroup$ @მამუკაჯიბლაძე What do you mean exactly by "resulting strings"? Aren't those exactly the (virtual) "almost horizontal" lines? Note that they are of limited length, either 22 (like 11,16,...,116) or 23 (like 74,79,...,184). And why "invariably"? It is exactly like, say, the strings mod 6, e.g. 6,12,...,42 (of length 7, as most of them) or 216,222,...,246 (length 6, there are very few like that). $\endgroup$ Commented May 26, 2017 at 18:59
  • $\begingroup$ Hmmm I see your point. I have hard time trying to formulate rigorously what I mean, but it is something like that: for each step size $s$, there is sort of characteristic "change rate" $\delta(s)$ - average difference between the imaginary parts of $P_n$ and $P_{n+s}$, not counting the "jumps". The first thing I cannot answer here is whether this average difference tends to zero or to some definite limit. Provided it has different values for different $s$, we can ask which $s$ minimizes the quantity $\delta(s)$; but it must be not quite right: $\endgroup$ Commented May 26, 2017 at 19:22
  • $\begingroup$ - since $\delta(s)$ can be arbitrarily small for appropriate (very large) $s$. It then seems to make sense "weighing" $\delta(s)$ and considering instead, say, $s\delta(s)$. Now I have not checked this but it looks like $5\delta(5)$ is smallest among all $s\delta(s)$ (well, probably excluding $1\delta(1)$). $\endgroup$ Commented May 26, 2017 at 19:28
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    $\begingroup$ @მამუკაჯიბლაძე I agree, but is it always clear how to define a jump? Try s=8. $\endgroup$ Commented May 26, 2017 at 20:39

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