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 $(X, {\mathcal A},\nu,\tau)$ and any $f\in L^p(\nu)$, $p>1$, the limit $$ \lim_{n\to \infty}{1\over \sum_{k=1}^{n} d(k)} \sum_{k=1}^{n} d(k)f(\tau^k x)$$ exists $\nu$-almost everywhere. We also obtain similar results for other arithmetical functions, like $\theta(n)$ function counting the number of squarefree divisors of $n$ and the generalized Euler totient function $J_s(n)$, $s>0$. We use Bourgain's method, namely the circle method based on the shift model.