This takes all the imaginary parts and puts them into bins of width 3. If x is only taken up to 1000 or 2000 rather than 5000 the same basic pattern emerges. There is some fluctuation in the later part, and surely this is due to the n=1 curve (roughly) y = |t|log(Sqrt[X/4*Pi]/Sqrt[2]), and the n=2 curve etc.

A bit more surprising in this histogram: lower imaginary parts seem a little less likely, with the rest of the range (roughly) from 0 to |t|*log(Sqrt[X/4*Pi]) reasonably equidistributed apart from the already mentioned fluctuations.

If we define N_t(X,Y) to be the number of zeros of H_t(z) with real part in [0,X] and imaginary part in [-Y,Y], this also becomes pretty revealing to plot, using the same data set.

Plotting 1/2* N_t(X,Y) / (X/4Pi) for various Y cutoffs (and X equal to 5x) gives the following plots:

No Y cutoff (just the Riemann-von Mangoldt formula for negative t)

Considering counts in aggregate rather than in small bins should have the effect of smoothing over the the fluctuations in the first histogram above. To make a somewhat wild guess (and ignore all second order terms), it may be that for fixed negative t,

N_t(X,Y) << (X/4Pi)*Min(log X, (2/|t|)*Y)

for Y in any fixed range [epsilon,Infinity), and to make a second guess this is an asymptotic formula as long as Y is growing. If Y is some fixed constant (not zero), this may still be true as an asymptotic formula, but at least judging by the histogram there could be some sort of term that changes the asymptotic a little. If one wanted to use numerics to test this quantitatively rather than qualitatively, it would probably be necessary to consider second order terms (e.g. log X should be replaced by log(X/4Pi), but there are certainly other missing terms like this too and I don't have a conjecture).

]]>The question amounts to understanding the winding number of for fixed choices of in some dyadic range . This amounts to understanding the winding number of an exponential sum that is roughly of the form times some well understood factors (there is a factor that is effectively localising to the region ). Comparing this against the sum that would come out of the Riemann-Siegel formula, one could indeed imagine that the winding number of the latter is only times that of the former, which would be consistent with your prediction, but this is _extremely_ non-rigorous :) It’s possible that one could use the theory of random Dirichlet series as a model to predict what should happen though.

]]>This was done using mathematica. Unfortunately the pattern is not so discernible. In blue is the exponential prediction. It may be that the data converges to this pattern very slowly, but there does seem to be a rather long tail…

(The large number of spacings in the very last bin all have size around 18.13; at first I thought this was a mistake, but this is the spacing between zeros on the first n=1 curve, and is as large a gap as one can have. Note that 4*Pi/log(2) is roughly 18.13, so this is in excellent agreement with the Remark 2.1 in the sharkfin ms the polymath prepared.)

By comparison, here is a histogram of (unrenormalized) spacings between zeros of H_0, and the Montgomery-Odlyzko prediction (I took the same number of zeros which ends up being up to just x = 3500). The convergence is fairly fast.

It could also be that spacings between real parts of H_t is not quite the ‘right’ object to look at, since zeros won’t follow a straight trajectory down to the real axis. (I am not sure if there is a rough description of the trajectory, up to some oscillating error term?)

]]>We now know that for increasingly negative , all complex zeros are ‘peeled off’ from the real line and eventually organise themselves on curves that can be explained by the dominance of two adjacent terms from an asymptotic (Dirichlet-type) series for . The most outer curve only requires the first two terms from this series and including more terms yields more curved patterns that are increasingly closer to the real line, however contain less complex zeros and oscillate more. These oscillations fade out when going further back in time as the curves move away from the real line.

All complex zeros originate from collisions of a pair of real zeros at a certain , So, the wave forms of the complex zeros near the real line should also ‘cast a shadow’ on the collision patterns of the real zeros at . As gets closer to , this ‘peeling off process’ clearly becomes increasingly more complex, however moving further away from things seem to become ‘quieter’. This culminated in the recent discovery that the final ‘peeling off-wave’ can be fully explained by the collisions of the real zeros of an Airy function (with the collisions occurring on the left edges of ‘sharkfin’ shaped structures). To explain this ‘most outer’ pattern at , we again only require the first two dominant terms of an asymptotic series (eq. 1.41 + 1.55), i.e. for the half integer lines and inducing the required oscillations in the sharkfin-shapes.

This is a long shot, but similar to the mechanisms that explained the oscillations in the complex zeros, could it be that subsequently adding Airy terms e.g. will fit a few of the noisier collisions patterns of real zeros at less negative ?

P.S.:

Surprisingly there even exists an Airy-Zeta function that seems to share a few high level properties with the Riemann Zeta-function.

*[This will be reworded in the next build of the pdf. -T]*

From the toy approximation (relying entirely on the “B” series) we heuristically have an approximation

of as a weighted exponential sum times some simple additional terms, where and (the wiki says but I think the latter is actually a slightly better approximation). When , the weights simplifies to which makes all of the phases coming from a single dyadic range of have equal “strength” as far as square root cancellation heuristics are concerned. But as one varies and , the weight changes shape and effectively localises the sum to much smaller ranges of (e.g., near 1, or near ), at which point the behaviour of the sum can be simplified to the point where one can understand the zeroes asymptotically.

]]>All the equations derived so far strongly indicate that information about always originates from and there is no information flowing vice versa. The case, i.e. the Riemann -function, seems therefore special. On the other hand, this is the original function that we chose to perturb in the and directions.

The recent insights from exploring the domain have changed my initial mental model in which real and complex zeros were flowing from a gaseous towards a solid state over time. It now more feels like represents a sort of thermodynamical equilibrium state (with the highest possible entropy) that is being disturbed by injecting an increasingly positive or negative and eventually converges the zero trajectories towards the regular (lower entropy) patterns that we can now all fully explain.

From the perspective of a model with at the centre, I find it still surprising that there isn’t at least some symmetry found between and . An obvious reason for the asymmetry probably is the dominant presence of complex zeros in the domain, however I could think of these two high level symmetries/similarities:

+t = real zeros always repel each other, complex zeros always attract each other.

0 = ‘peaceful’ coexistence between all zeros.

-t = real zeros always attract each other, complex zeros always repel each other.

+t = there is a minimum speed by which complex zeros fall towards their conjugates and collide.

0 = all zeros in a standstill.

-t = there is a minimum speed by which real zeros fall towards a partner and collide.

For the latter similarity we know that this minimal speed is determined by a solution to an ODE, respectively the DBN upper bound for complex zeros at (using the known supremum ) and the Airy-function for some ‘latest collision’ real zeros at larger .

So, we now know how far imaginary zeros can penetrate the domain (the DBN bound, or hey, hey: $\Lambda \le 0.22$ !) and we also know how far real zeros can penetrate the domain (the curve drawn through the tips of the sharkfins, thereby ignoring the zeros on the straight lines at half integers).

The bound for the complex zeros was derived from their dynamics, so from a symmetry perspective, could a bound for the real zeros at also be derived from their dynamics? I.e. when one starts from a set of non-trivial zeros up to at , is it possible to derive from their (assuming only attractive) dynamics the (x, -t) upper bound before which all these zeros must have collided?

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