Let $f_i(x1,\ldots,x_m)$ be polynomials in $m$ variables over $\mathbb{Z}_p$, for $i=1,\ldots, r$. Let $N_n$ be the number of elements in $\{x\mod p^n\mid x\in\mathbb{Z}_p^m, f_i(x)=0\text{ for }i=1,\ldots,r\}$.
In Jan Denef's paper on The rationality of the Poincaré series associated to the $p$-adic points on a variety in Lemma $3.1$ he uses the following integral $$\int\limits_{(x,p^n)\in D}\lvert dx\rvert$$ where $\lvert dx\rvert$ denotes the Haar measure on $\mathbb{Z}_p^m$. Also $$D= \{(x,w)\in\mathbb{Z}_p^m\times \mathbb{Z}_p\mid \exists y\in\mathbb{Z}_p^m:x\equiv y\mod w,\text{ and }f_i(y)=0,\text{ for }i=1,\ldots,r\}$$
So what it comes down to is calculating the haar measure of $$\{x\in\mathbb{Z}_p^m\mid \exists y\in\mathbb{Z}_p^m:x\equiv y\mod p^n,\text{ and }f_i(y)=0,\text{ for }i=1,\ldots,r\}$$
I think the result should be $\dfrac{N_n}{p^{nm}}(1-\dfrac{1}{p})$. Does anyone know how to do this calculation?
Edit: the result should be $\dfrac{N_n}{p^{nm}}$.