The convergence of some weighted means of values of a function

Abstract

Consider a sequence $(a_n)_{n\ge 1}$ of complex numbers such that for some positive number p we have $\lim _{n\rightarrow \infty }\frac{1}{n^p}\left( \sum _{k=1}^na_k\right) =\ell \in \mathbb {C}$. We prove that, under some mild conditions on the sequence (positivity or absolute boundedness in the mean, etc.), we have

$$\begin{aligned} \lim _{n\rightarrow \infty } \frac{1}{n^{q}}\sum _{k=1}^nk^{q-p}a_kf\left( \frac{k}{n}\right) = p\ell \int _0^1x^{q-1}f(x)dx \end{aligned}$$

for all $q>0$ and all Riemann integrable functions $f:[0,1]\rightarrow \mathbb {C}$. Some applications to probability theory and to number theory are also discussed.