## 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.

### MSC:

 11J71 Distribution modulo one

### Keywords:

bins; histogram; partial sums
Full Text:

### References:

 [1] Beck J., Ann. of Math. (2) 140 (2) pp 449– (1994) · Zbl 0820.11045 [2] Hardy G. H., Abh. Math. Sem. Hamburg 1 pp 212– (1922) [3] Khintchine A., Math. Ann. 92 pp 115– (1924) · JFM 50.0125.01 [4] Montgomery H., Ten lectures on the interface between analytic number theory and harmonic analysis (1994) · Zbl 0814.11001 [5] Ostrowski A., Abh. Math. Sem. Hamburg 1 pp 77– (1922)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.