Just as I thought that Polymath5 was perhaps beginning to lose momentum, it has received a sudden boost, in the form of this comment of Terry’s, which has potentially completed one of the hoped-for steps of the proof strategy, by establishing that if there is a counterexample to EDP, then there is a “moral counterexample” that is completely multiplicative.
There are a few points that need clarifying here (which I am just repeating from Terry’s comment). The first is that the multiplicative counterexample is over the complex numbers: that is, it is a sequence with values in . The second is that instead of having bounded partial sums (and hence bounded discrepancy), it has mean-square partial sums with bounded lim inf. That is, there exists a constant such that infinitely often. Turning this round, in order to prove EDP it is sufficient to prove that for every completely multiplicative function we have the inequality for some function that tends to infinity. (more…)