Maximum mean discrepancies of Farey sequences

TL;DR

This work connects a polynomial-rate convergence of the maximum mean discrepancy between the uniform distribution on $[0,1]$ and Farey-sequence empirical measures to the Riemann hypothesis, for a broad class of kernels including Matérn kernels with $\nu\ge 1/2$. By embedding the problem in RKHS theory and Sobolev regularity, it identifies precise conditions under which RH is equivalent to $\mathrm{MMD}(F_n)=O(n^{-3/2+\varepsilon})$ (or $O(N^{-3/4+\varepsilon})$). It also analyzes energy-distance kernels, deriving explicit MMD formulas and showing that RH corresponds to cancellations between main components of the MMD, thereby linking number-theoretic distribution properties to kernel-based statistics. The results provide a framework where classical Farey-discrepancy phenomena are interpreted through kernel methods and RKHS structure, with implications for roughness-regularized discrepancy measures and kernel choices in statistical testing.

Abstract

We identify a large class of positive-semidefinite kernels for which a certain polynomial rate of convergence of maximum mean discrepancies of Farey sequences is equivalent to the Riemann hypothesis. This class includes all Matérn kernels of order at least one-half.

Figures (2)

• Figure 1: The first 18 Farey sequences $F_1, \ldots, F_{18}$.
• Figure 2: The plots show $\mathrm{MMD}(F_n) \cdot n^{3/2}$, the normalised MMDs of Farey sequences, up to $n = 250$ for Matérn kernels with (i) $\nu = 1/2$ and $\lambda = 1$, (ii) $\nu = 3/2$ and $\lambda = 3^{1/2}$, and (iii) $\nu = 5/2$ and $\lambda = 5^{1/2}$. Note that $n$ is the index of the Farey sequence, not the number of points. For $n = 250$ we have $N = \lvert F_n \rvert = 19,\!025$.

Theorems & Definitions (10)

• theorem 2.1
• Proposition 2.2
• remark 2.3
• remark 2.4
• lemma 4.1
• proof
• proof : Proof of Theorem \ref{['Thm:Main']}
• lemma 4.2
• proof
• proof : Proof of Proposition \ref{['Cor:Main']} for energy-distance kernels with $\alpha \in [1, 2)$] Assumption (b) holds by setting $m = 2$ in Lemma \ref{['lemma:energy-distance-rkhs']}. For $\alpha = 1$ and $x, y \geq 0$ the energy-distance kernel becomes $K_1(x, y) = \lvert x \rvert + \lvert y \rvert - \lvert x - y \rvert = \min\{x, y\} .$ In the proof of Theorem \ref{['Thm:Main']} we noted that the RKHS of $K(x, y) = 1 + \min\{x, y\}$ on $[0, 1]$ is norm-equivalent to $W^{1,2}([0,1])$. The RKHS of $K$ consists of sums of constant functions with elements of the RKHS of $K_1$ Berlinet2004. From the characterisation of $H_\alpha(\mathbb{R})$ in BartonPoor1988 that we used in the proof of Lemma \ref{['lemma:energy-distance-rkhs']} it also follows that $H_\alpha(\mathbb{R}) \subseteq H_\gamma(\mathbb{R})$ if $\alpha \geq \gamma$. This inclusion is inherited by RKHSs on $[0, 1]$. Therefore $K_\alpha$ satisfies assumption (a) for $\alpha \in [1, 2)$. TK was supported by the Research Council of Finland projects 338567 ("Scalable, adaptive and reliable probabilistic integration"), 359183 ("Flagship of Advanced Mathematics for Sensing, Imaging and Modelling"), and 368086 ("Inference and approximation under misspecification"). R. J. Adler and J. E. Taylor. Random Fields and Geometry. Springer Monographs in Mathematics. Springer, 2007.R. J. Barton and H. V. Poor. Signal detection in fractional Gaussian noise. IEEE Trans. Autom. Control, 34(5):943--959, 1988.A. Berlinet and C. Thomas-Agnan. Reproducing Kernel Hilbert Spaces in Probability and Statistics. Springer, 2004.C. Cobeli and A. Zaharescu. The Haros-Farey sequence at two hundred years. Acta Univ. Apulensis. Math. - Inf., 5:1--38, 2003.J. Dick and F. Pillichshammer. Discrepancy theory and quasi-Monte Carlo integration. In A Panorama of Discrepancy Theory, pages 539--619. Springer Cham, 2014.F. Dress. Discrépance des suites de Farey. J. Théor. Nr. Bordx, 11(2):345--367, 1999.A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi. Tables of Integral Transforms, volume 1. McGraw-Hill, 1954.J. Franel. Les suites de Farey et le problème des nombres premiers. Nachr. Ges. Wiss. Göttingen, pages 198--201, 1924.A. Gretton, K. M. Borgwardt, M. J. Rasch, B. Schölkopf, and A. Smola. A kernel two-sample test. J. Mach. Learn. Res., 13:723--773, 2012.G. H. Hardy and E. M. Wright. An Introduction to the Theory of Numbers. Oxford University Press, 4th edition, 1960.S. Kanemitsu and M. Yoshimoto. Euler products, Farey series and the Riemann hypothesis. Publ. Math. Debrecen, 56(3--4):431--449, 2000.M. Mikolás. Farey series and their connection with the prime number problem. I. Acta Sci. Math. (Szeged), 13:93--117, 1949.M. Mikolás. Farey series and their connection with the prime number problem. II. Acta Sci. Math. (Szeged), 14:5--21, 1951.M. Mikolás and K. Sato. On the asymptotic behaviour of Franel's sum and the Riemann hypothesis. Results Math., 21:368--378, 1992.H. Q. Minh. Some properties of Gaussian reproducing kernel Hilbert spaces and their implications for function approximation and learning theory. Constr. Approx., 32(2):307--338, 2010.H. Niederreiter. The distribution of Farey points. Math. Ann., 201:341--345, 1973.E. Novak and H. Woźniakowski. Tractability of Multivariate Problems. Volume II: Standard Information for Functionals. Number 12 in EMS Tracts in Mathematics. European Mathematical Society, 2010.V. I. Paulsen and M. Raghupathi. An Introduction to the Theory of Reproducing Kernel Hilbert Spaces. Number 152 in Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2016.D. Sejdinovic, B. Sriperumbudur, A. Gretton, and K. Fukumizu. Equivalence of distance-based and RKHS-based statistics in hypothesis testing. Ann. Stat., 41(5):2263--2291, 2013.M. L. Stein. Interpolation of Spatial Data: Some Theory for Kriging. Springer Series in Statistics. Springer, 1999.A. W. van der Vaart and J. H. van Zanten. Reproducing kernel Hilbert spaces of Gaussian priors. In Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh, pages 200--222. Institute of Mathematical Statistics, 2008.T. T. Warnock. Computational investigations of low-discrepancy point sets. In Applications of Number Theory to Numerical Analysis, pages 319--343. Elsevier, 1972.H. Wendland. Scattered Data Approximation. Number 17 in Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, 2005.M. Yoshimoto. Farey series and the Riemann hypothesis. II. Acta Math. Hungar., 78(4):287--304, 1998.B. Zwicknagl and R. Schaback. Interpolation and approximation in Taylor spaces. J. Approx. Theory, 171:65--83, 2013.