This is another post for the sake of not having too many comments on any single post. I actually wrote a completely different one yesterday, but it contained some theoretical thoughts that are probably better held back until the project starts in earnest. So I’m going to make this post very short indeed.
A quick report on what’s going on right at the moment. We have just been fed some more data: several more sequences of length 1124 and discrepancy 2, some sequences that have HAP-sums bounded between numbers like -1 and 3 instead of -2 and 2, and some nice two-dimensional examples. The Polymath5 wiki has some details about these and other aspects of the problem.
If anyone feels like doing an experiment that I’d be interested to know the result of, they could investigate the following sequence. I want to know whether it is good for anything. It’s the simplest example I can think of of a sequence that appears to exhibit multiplicative behaviour but keeps breaking the pattern. Let be the completely multiplicative function that takes every prime to , where . Then define to be 1 or -1 according to whether the imaginary part of is positive or negative. (Of course, .) At what kind of rate does the discrepancy of this sequence seem to grow?
Another experimental question: are all the HAP subsequences of the new 1124 examples significantly different from all the HAP subsequences of the old one? In other words, is it reasonable to say that they are completely different examples, or are they more like modifications of HAP subsequences of the original one?