Partial sums of \(\zeta(\frac 12)\) modulo 1. (English) Zbl 1006.11037

Let \(P_{s}(n)=\sum^{n}_{j=1}j^{-s}.\) For fixed \(s\) near \(s=\frac{1}{2},\) the unit interval is divided into bins and a histogram is constructed by counting how many of the partial sums \(P_{s}(1), P_{s}(2), \ldots, P_{s}(N)\) lie in each bin mod 1. The properties of the histogram are predicted by a random model unless \(s=\frac{1}{2}\). When \(s= \frac{1}{2}\) the histogram is surprisingly flat, but has a few strong spikes. To explain the surprises at \(s=\frac{1}{2}\), classical results about Diophantine approximation, lattice points, and uniform distribution of sequences are used.


11J71 Distribution modulo one
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