I do not have too much to say about the new programme to use SDP to solve EDP, other than that it is still being actively pursued. One remark I would make is that I have rather forgotten recently about keeping the wiki updated, and SDP is a major omission from it. It would be good to have a theoretical account of the method (which I suppose I could paste quite easily from one of my existing posts), and an experimental page devoted particularly to data connected with this approach, with links to all the matrices, sequences and plots that have been produced.
So far, the data has had some expected features (such as the sequence values tending to be larger when is smooth) and some puzzling ones (such as the fact that is much much larger than ). An initial hope for the method was that the experimental data would give rise to a very precise conjecture, but so far that has not happened. There are, however, various avenues that have not been fully explored, and I still have some hope that we will suddenly stumble on some data that we can fully understand.
One way of doing this is to introduce a smoothing. One idea I had turned out to go too far: in the search for a pretty formula I ended up with a version of EDP that was false. For the record, here is that version. One can think of the discrepancy of a sequence as its largest inner product with the characteristic function of any HAP. Those characteristic functions have sharp cutoffs (in that they go up to for some and then suddenly stop), so one might hope to make the problem nicer by smoothing the cutoffs. One natural way of doing this is to look at functions such as , which take the value at and 0 at non-multiples of . For this to be a sensible idea, we need a “smoothed EDP” to be true. That is, we need it to be the case that for every sequence there exist and such that has absolute value at least . However, this is false for the character-like function . The multiplicativity of implies that it will be enough to show this when , so let us fix some and calculate the sum .
We do this in the usual way. First, let us look at non-multiples of 3. That gives us the sum
which equals . More generally, the sum over multiples of that are not multiples of works out as . One can check that the function is increasing in the interval , so if we sum this over all then the alternating series test tells us that the total is at most 1/3, whatever the value of .
A similar argument works if we use a weight of instead, but the calculations are easier. By the alternating series test, we know that is positive and at most 1. Call this value . Then , which is at most .
At one point I observed that if you have a sequence of bounded discrepancy then the Dirichlet series is uniformly bounded for all positive real , and wondered if we could deduce anything useful from this. The fact that has this property shows that we certainly cannot derive a contradiction, though it leaves open the possibility of deducing that the sequence is forced to have character-like behaviour.
In the light of these observations, what hope is there for obtaining nicer formulae by smoothing? The answer is that even if we cannot smooth the HAPs, we can at least smooth the interval of integers we are thinking about as a whole. That is, instead of thinking of discrepancy as a function of (the length of the sequence ), we could take an infinite sequence , associate with a weight , and think of the discrepancy as a function of the decay rate (or equivalently the half-life, which is proportional to ). That is, we would like to prove that for each there must exist such that has absolute value at least , where as .
If we try to prove this using Moses’s SDP approach, then, as Moses points out, there is a nice formulation of the problem. We would like to find non-negative coefficients that sum to 1, and coefficients such that , such that
for every sequence . This is similar to what we were looking at before, but now we have attached some slowly decaying weights to the .
Note that we could think of the previous version of the problem as doing exactly the same, but with weights that equal 1 up to and thereafter. I am hoping that with these smoother weights, we will get nicer numbers coming out.
I would like to end this post by drawing attention to Gil’s polynomial method. This is a completely different approach to EDP, where one constructs a polynomial over a finite field that is identically zero if and only if EDP is true. Rather than explain the idea in detail, let me link to two comments in which Gil himself gives a nice explanation. He introduced the idea in this comment and an interesting variant of the idea in this one. It would be good to add this method too to the list of possible proof techniques on the wiki.