We now understand what happens to the real zeros when , and also what happens to the complex zeros when (all gone when ). However we haven’t yet fully explored what happens to these curves of complex zeros when .

Maybe this is trivial, but when I compute the complex solutions for , it seems at first that the curves just drift further away from the real line and then ‘straighten out’ into horizontal lines at increasingly negative . However at a certain point new complex zeros start to emerge close to the origin as can be seen in this plot ( with is the most inner curve):

*[It appears that if the link ends in something like .jpg then it is automatically rendered, but if it is not a recognised image extension then it is not. -T]*

The complex zeros appear to originate from a certain and then travel into the domain when decreases further (note that the zeros of are not conjugated pairs, e.g. at the curves would become straight lines at . I guess a slightly different version for Z() exists that produces the conjugated version of the curves).

Note that the outer right sides of the n-curves in the plot are all relatively close to the vertical lines of real zeros (the zeros always stay on the left side of these vertical lines). Where could these new complex zeros come from and how could they travel through the real line without a collision with their conjugate and thereby inducing a new pair of real zeros?

]]>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]*