Corrections to figures

[teorth]

I guess I should use this issues forum rather than email for requests for corrections to the figures. Here's what I have so far:

• For the left figure in Figure 2, the horizontal axis should be log_{10} x rather than 10^x.

• For figure 5, it should be |f_t + RS| rather than |f_t + RS|. Actually I've just tweaked the notation in the paper so that one can use the more precise description |f_t - C_t/B_t|.

• To complement Figures 3,4, it would also be nice to have a figure displaying what happens at significantly larger values of x (e.g. x ~ 10^5 or x ~ 10^6, if numerically feasible) to illustrate how the accuracy of the f_t approximation to H_t/B_t improves with x.

[km-git-acc]

@teorth @rudolph-git-acc I have fixed the labeling in Figs 2 and 5 and uploaded the edited figures and files to the writeup folder. We will also check on the 3rd point. I think the bottleneck was that H_t is slow to compute at large x (for eg. 5 sec for x near 10^5 and 90 sec for x near 10^6) so we went for the easier option, but atleast the x near 10^5 graph could be done in a few hours.

[rudolph-git-acc]

@teorth @km-git-acc

Oops. It seems we did some work in parallel here and was just planning to post the corrected graphs as well. Have managed to get plots for x~10^6. Just waiting for the final one to complete. Will post shortly.

[rudolph-git-acc]

@teorth @km-git-acc

Below are all updated graphs for inspection. In case the 'two in a row' graphs become too small to read in the PDF, we could also decide to place them vertically.

Figure 2: now with correctly labelled x-axis.

Figure 5: using the more precise description |f_t - C_t/B_t|.

Figures 3,4: with additional graphs for the x ~10^6 domain.

[teorth]

These look great, thanks!

I may have set up an edit conflict as I was also editing the intro.tex file without first merging from master; I am not sure how to fix it from my end but perhaps the merge can be completed from the master end.

Looking at the wide fluctuation of H_t/B_t I wonder if it is also worth having a separate plot comparing H_t/B_t with the partial Euler products that we use to select the location of the barrier, perhaps re-using some of the ranges of x,y,t in one of the above plots for sake of comparison. Certainly these figures make it clear that the choice of barrier location can be important!

[rudolph-git-acc]

Yes, visualizing the impact of the partial Euler product is a great idea! We'll work this further.

To avoid editing conflicts, it is probably easiest when you simply ignore the recent changes we made and just finish your detailed walkthrough of the write-up. We can then easily re-include all the images and tables at the end of the process (and also ensure the formatting and positioning works correctly).

[km-git-acc]

Have resolved the merge conflict, and also updated the pdf with the above figures. Rudolph's suggestion is a good one, in that we can now wait for all the tex edits to finish first.

[km-git-acc]

@teorth @rudolph-git-acc

Below are two graphs comparing the f_t approximation with partial euler products (t=0,y=1) upto the first 30 primes (the first one near X=6x10^10+83951.5, and the second one near X=10^6). As the number of primes in the euler product increases, the fit w.r.t f_t improves significantly as expected.

Is this in the right direction, and are there any pointers to bring the plot style/formatting more in line with the paper?

[teorth]

Looks good! For the first figure, can you make the offset around 83951.5 rather than 83935.5 (so that the offset will take on both negative and positive values) to emphasise the spike at 83951.5?

Also, what exactly is p1, p2, p5, etc.? The current writeup uses the notation eulerprod() instead (presumably this corresponds to p_27, though actually I was a bit curious as to why the composite number 27 was chosen as the cutoff). Of course on the figure we need to abbreviate the notation anyway, but this can presumably all be explained in the figure caption.

If you could add the figure to the end of Section 7 (dirichlet.tex) that would be great (I am not currently editing the files, so there should be little chance of any edit conflicts).

[km-git-acc]

I made the changes and a few others, and have added the below chart to the writeup.

Exact figure placement with Latex seems a bit tricky, and currently the chart is appearing on pg. 37 after Section 8 starts. p1,p2,.. labels have been replaced by ep_n(x) = eulerprod(x,p_n)

Also, the number 27 used in the eulerprod definition is actually 29, since the first 10 primes were used for calculations. Have changed that as well.

[teorth]

Looks great, thanks! Will resume going through the writeup now.

[teorth]

Here's another possibility for a figure in the introduction: a comparision between the error |f_t - H_t/B_t| and the error bound e_A + eB + e{C,0} (as well as the component factors e_A, eB, e{C,0}). Probably the most informative graphic will have x vary over a long range, e.g. using log-scale for x, then the error should oscillate wildly (similar to the left figure in Figure 2) but the error bounds e_A, eB, e{C,0} should be much smoother. (But one could also have a closeup picture near a place where N gets incremented to highlight the discontinuity present there.)

[rudolph-git-acc]

@teorth @km-git-acc

Yes, such visuals about the error terms are definitely doable. I will start working these!

[teorth]

Thanks! Another image request: the asymptotic analysis in Theorem 1.5 suggests that the zeroes of H_t eventually approach the points where \frac{X}{4\pi} \log \frac{X}{4\pi} - \frac{X}{4\pi} + \frac{11}{8} + \frac{t}{16} \log \frac{X}{4\pi} is an integer. I don't know if you have the capability to locate zeroes of H_t for large enough x that one can see this "solidifying" of the zeroes, but it would be nice to have a graphic at large x showing this (and perhaps also a graphic at small x showing the more disordered behaviour of zeroes).

[rudolph-git-acc]

@teorth @km-git-acc

EDIT: Made progress on visualising the error bounds. Below are the first versions. The discontinuaties at the N-jumps are already clearly visible and it also doesn't seem to make much sense from a visual perspective to move x much further beyond 5. Bit surprised to see that the error |f_t-H_t/B_t| isn't actually "oscillating wildly" and rather seems to follow a quite decently fluctuating pattern. The graphs below show that e_B stays above e_A and both are much smaller than the dominating term e_C,0.

We have developed a real-root finder that works well up till at least the x=10^12 domain (it is based on the ABC_eff proxy for Ht). Just ran a few tests and the integer-effect is very clearly there. It gets stronger with increasing x and also with increasing t.

EDIT: here is a first attempt to show how zeros x_i "solidify" over time at integer values of g(x_i,t)=\frac{x_i}{4\pi} \log \frac{x_i}{4\pi} - \frac{x_i}{4\pi} + \frac{11}{8} + \frac{t}{16} \log \frac{x_i}{4\pi}. Need to think a bit more about how to visualise the "solidification" for a frozen time with an increasing x. Will work that further tomorrow.

EDIT: below is a graph showing how for a frozen time t=0.5, the zeros of H_t will 'solidify' at integer values of g(x_i,t) when x increases. Using a log scale for x, I calculated the first zeros for each step in the exponent of the power of 10 and then established for each of these zeros the distance to their nearest integer. As expected; for t's > 0.5 the convergence to zero (=integer) happens increasingly faster.

EDIT: Did run a similar graph now running up to x=10^16.

[rudolph-git-acc]

@teorth @km-git-acc

Quick update: during the last two weeks, KM and I have been progressing the numerical work through using grid-computing (Boinc) to conditionally proof a DBN <= 0.11 and <= 0.10. After being listed on the Boinc-homepage, we saw a massive increase in the number of volunteering participants, with some bringing extremely powerful computers to this great collaborative effort.

Both DBN-cases have a Triangle bound that is already positive at the Barrier location, so no further numerical work is required beyond the verification that no zeros have passed the Barrier itself. We probably need less than a week to fully complete both.

Please let us know whether any additional visuals/tables/texts need to be produced for the write-up on top of the ones in the previous posts.

[teorth]

Great! I managed to miss being notified of your previous update, but the error bound pictures look great, feel free to insert them into the writeup (I can edit them later if needed).

Probably the reason that f_t - H_t / B_t is not too wild is that it is being dominated by the "C" term (compare for instance the oscillations with the plot of C_0 in Figure 6 of the writeup.

One other graphic that came to mind relates to Figure 2 of the writeup. In addition to a plot of the relative magnitude |H_t|/|B_t| vs log x on the left of Figure 2, one could also add a plot of log |H_t| and log |B_t| vs log x to illustrate how B_t captures the bulk of the exponential decay of H_t. (Also it illustrates how more naive ways to calculate H_t would need extremely high numerical precision at large x due to how small H_t is.)

[rudolph-git-acc]

@teorth @km-git-acc

Below is the log|Ht| and log|Bt| graph. If ok, I will include this together with the error-term graphs in the write-up later today. Shall I also include the 'solidification' graphs from the previous post?

[teorth]

Looks great! Please do include the previous figures also.

[rudolph-git-acc]

@teorth @km-git-acc

All done :) Not all graphs are well positioned yet, but probably best to wait until the write-up is complete and then 'force' them into the right location.

[km-git-acc]

@teorth @rudolph-git-acc As an update, recently using the grid computing setup, we computed winding numbers in the barrier region, X=9*10^21 + [70686,70687], y=[0.11832,1], t = [0,0.093], and all the winding numbers turned to be zero (around 60000 t-steps), (https://github.com/km-git-acc/dbn_upper_bound/blob/master/output/windingnumbers/windnum_nolemma_x9000000000000000070686_y_0.11832_1_t_0_0.093.txt) giving dbn < 0.1 conditionally.

[teorth]

Hi, sorry for not being around the past few weeks, it's been crazily busy around here.

I'm going through the writeup again, hopefully to get it to a near-final form (even if the numerics keep going, I think activity on the theory side has basically ceased and so it would be good to wrap things up). I'm looking again at Figure 3 (the last figure Rudolph provided here) and wonder if it may look better if we plot log|H_t| and log |B_t| against x rather than log x, for the same range of x (1 to 10^7). Then the plot should be roughly linear. Either way it does convey though that H_t gets incredibly tiny for large x!

[rudolph-git-acc]

Here is an update version indeed showing an almost linear relation. Happy to make any other changes.

[teorth]

Thanks! I have incorporated this figure into the latest writeup. On a related note, it occurs to me that for Figure 2, a plot of |H_t/B_t - 1| is slightly more informative than a plot of |H_t/B_t| since the latter does not preclude the scenario in which H_t differs from B_t by a large phase (e.g. for instance if H_t was close to -B_t). One could in principle also do this for Figure 4, 5 except that for the right-hand figure the residual H_t/B_t - f_t would presumably be visually indistinguishable from zero (and one would have to rescale the axes), and we do something like this anyway for Figure 6, so perhaps we could leave those as is.

[rudolph-git-acc]

Below is the new version of Figure 2 now with |H_t/B_t - 1| :

[teorth]

Great! I note that the description of the right-hand image (both above the image and in the box) still has |H_t|/|B_t| rather than |H_t/B_t-1|, is this a typo?

[rudolph-git-acc]

Ah, not a typo. I had actually fixed it for the left hand graph only. Will do both!

[rudolph-git-acc]

Here is the updated version, now also done the right-hand graph:

[teorth]

Great! I now put this into the writeup.

[teorth]

Finally got some time to work on the writeup again. I think there may be a typo in what is currently Figure 9 (Integerconvergence_t_and_xdirection.png) on the writeup, in that the x-axis for the left figure should be t rather than log{10} x. The original version of the figure (the third picture in Rudolph's Sep 25 comment) has the axis labeled correctly though. Would it be possible to fix the axis labeling? Thanks.

[rudolph-git-acc]

Good catch! This was the result of a stupid copy/paste error. Here is the updated version. Please let us know how we can further help with the write-up.