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2001 Partial Sums of $\zeta(\half)$ Modulo 1
Jade Vinson
Experiment. Math. 10(3): 337-344 (2001).

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.

Citation

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Jade Vinson. "Partial Sums of $\zeta(\half)$ Modulo 1." Experiment. Math. 10 (3) 337 - 344, 2001.

Information

Published: 2001
First available in Project Euclid: 25 November 2003

zbMATH: 1006.11037
MathSciNet: MR1917422

Rights: Copyright © 2001 A K Peters, Ltd.

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Vol.10 • No. 3 • 2001
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