# Integration over ideles over $\Bbb{Q}$ , Tates thesis special case

Let $f\in S(A_\Bbb{Q})$ that is $f$ is adelic Schwartz-Bruhat function over $\Bbb{Q}$, such that all its components in the finite places are characteristic functions of the corresponding ring of integers and the infinite component is denoted by $h$. $A_\Bbb{Q}$ is the adele of $\Bbb{Q}$

Let $Z(f,s)=\int_If(x)|x|^sd^{*}x$ where $I$ is the idele group of $\Bbb{Q}$. How to show that for $\sigma \in (0,1)$ where $\sigma$=Real($s$)

(for all $h$ which is a Schwartz function over $\Bbb{R}$, $Z(f,s)=0$) iff $\zeta_\Bbb{Q}(s)=0$.

Moreover how to show that

$Z_+(f,s)[=$ by definition $\int_{|x|>1}f(x)|x|^sd^{*}x$)]$=\int_{\Bbb{R}}h(x)(|x|^{s-1}\sum_{0<n<|x|+1}n^{-s})dx$

I tried by decomposing $Z(f,s)$ into local components but nothing worked out. I am just learning Tates thesis from the book Ramakrishnan Valenza.