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See here for Franel and Landau's equivalent forms of the Riemann hypothesis in terms of the uniformity of distribution of Farey sequences.


On one view, Farey sequences arise by projecting those lattice points in ${\Bbb Z}^2$ that fall within certain triangular regions to a certain line segment.

So certainly higher-dimensional analogues suggest themselves. For example one might project points $(x_1,\ldots,x_n)$ in ${\Bbb Z}^2$ with $m \geq x_1 \geq x_2 \geq \cdots \geq x_n \geq 0$ from the origin to the hyperplane $x_1=1$. Note that choosing a measure of uniformity of distribution seems less straightforward, or at least less canonical, in the higher-dimensional case.

My question: with the "right" generalization of Farey sequence and the "right" measure of uniformity of distribution, does one obtain new equivalent forms of the Riemann hypothesis or perhaps related conjectures?

(I mean to cast a wide net, so please feel free to offer answers that stretch my paradigm in a reasonable way, thanks.)

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I don't know. The measures of uniformity of distribution in the Franel/Landau formulas you refer to are the discrepancy and the $L^2$ discrepancy, I believe; the theory of higher-dimensional discrepancy is well-established, and discussed in Kuipers and Niederreiter (and elsewhere). –  Gerry Myerson Jan 25 '11 at 0:00

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