Partial Sums of $\zeta(\half)$ Modulo 1

Abstract

Let $P_s(n) = \sum_{j=1}^{n} j^{-s}$. For fixed $s$ near $s=\frac12$, we divide the unit interval into bins and count how many of the partial sums $P_s(1)$, $P_s(2)$, \dots, $P_s(N)$ lie in each bin $\mone$. The properties of the histogram are predicted by a random model unless $s=\frac12$. When $s=\frac12$ the histogram is surprisingly flat, but has a few strong spikes. To explain the surprises at $s=\frac12$, we use classical results about Diophantine approxmation, lattice points, and uniform distribution of sequences.