Ergodic theorems with arithmetical weights

Abstract

We prove that the divisor function d(n) counting the number of divisors of the integer n is a good weighting function for the pointwise ergodic theorem. For any measurable dynamical system (X, A, ν, τ) and any f ∈ L p (ν), p > 1, the limit $$\mathop {\lim }\limits_{n \to \infty } \frac{1}{{\Sigma _{k = 1}^nd\left( k \right)}}\sum\limits_{k = 1}^n {d\left( k \right)f\left( {{\tau ^k}x} \right)} $$ exists ν-almost everywhere. The proof is based on Bourgain’s method, namely the circle method based on the shift model. Using more elementary ideas we also obtain similar results for other arithmetical functions, like the θ(n) function counting the number of squarefree divisors of n and the generalized Euler totient function J s (n) = Σ d|n d s μ(n/d), s > 0.