Partial Sums of ζ(½) Modulo 1

Abstract

Let Ps(n) = Σn j=1j−s. For fixed s near s = ½, we divide the unit interval into bins and count how many of the partial sums Ps(1), Ps(2), …, Ps(N) lie in each bin mod 1. The properties of the histogram are predicted by a random model unless s = ½. When s = ½ the histogram is surprisingly flat, but has a few strong spikes. To explain the surprises at s = ½, we use classical results about Diophantine approxmation, lattice points, and uniform distribution of sequences.