Okay im trying to integrate the following over the padic numbers to get an euler factor for the completed riemann zeta function
$\int_{Q^x_{p}} 1_{Z^x_{p}} |x|^s d^xx$
where $d^xx$ is with respect to the multiplicative Haar Measure on the p-adics. It seems that when I integrate with respect to the multiplicative Haar Measure I don't get the correct result. Howver if I use the additive Haar measure I do get the desired result. But in his thesis Tate uses the multiplicative group. Here are my steps
$\int_{Q_{p}} 1_{Z^x_{p}} |x|^s d^xx = \int_{Z^x_{p}} |x|^s d^xx$
Then i split $Z^x_p$ into a disjoint union of products $p^n$ and $u \in Z^*_p$
$\sum^{\infty}_{n=1} \int_{p^n{Z_{p}}} |p^n u|^s \frac{du}{|p^n u|}$
$\sum^{\infty}_{n=1} \int_{p^n{Z_{p}}} |p^n|^{s-1} du$
$\sum^{\infty}_{n=1} p^{-n(s-1)}$
$\sum^{\infty}_{n=1} (\frac{1}{p^{s-1}})^n$
$\frac{1}{1-p^{s-1}}$
As you can see I get something similair to the euler factor but not quite there, and if I was to use the additive measure we would get the desired result. I think it is an issue with the scaling of $Z_p$ with the multiplicative measure but i can not pinpoint the error. Thanks for any help I can get.