Maximum mean discrepancies of Farey sequences

TL;DR

This paper explores the connection between the convergence rates of maximum mean discrepancies of Farey sequences and the Riemann hypothesis, identifying a broad class of kernels where this equivalence holds.

Contribution

It introduces a large class of positive-semidefinite kernels, including Matérn kernels of order at least one-half, linking their convergence rates to the Riemann hypothesis.

Findings

Convergence rate of MMD of Farey sequences is equivalent to the Riemann hypothesis for certain kernels.

Includes all Matérn kernels of order ≥ 0.5 in the class where this equivalence holds.

Establishes a new connection between kernel methods and number theory.

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 Mat\'ern kernels of order at least one-half.