Ergodic averages with the Hecke eigenvalue square weights and the Piltz divisor function weights

TL;DR

This paper demonstrates that Hecke eigenvalue squares and Piltz divisor functions serve as effective weights for the pointwise ergodic theorem, advancing understanding of their ergodic properties and addressing open problems.

Contribution

It establishes the ergodic theorem with these arithmetical weights and extends results to other arithmetical functions, partially solving open problems.

Findings

Hecke eigenvalue squares are good weights for ergodic averages.

Piltz divisor functions also serve as effective weights.

Results extend to various other arithmetical functions.

Abstract

In this paper, we prove that the Hecke eigenvalue square for a holomorphic cusp form and the Piltz divisor functions are good weighting functions for the pointwise ergodic theorem. This partially solves problems suggested by Cuny and Weber. Additionally, we prove similar results for various other arithmetical functions in the last section.