Im currently reading this paper about the Dedekind zeta function and I'm interested in its functional equation. Despite not being very proficient in algebraic number theory I believe everything is clear besides this one integral transform. Though it regards algebraic number theory, it's nothing more than the calculation of a Jacobian.
So we have two integers $r_1,r_2$ and $r=r_1+r_2 -1$ where $r$ is the rank of $\mathcal{O}^x_K$ and $r_1+2r_2=n=[K:Q]$ is the dimension of the field over the rationals. The integral is of the form $\int_{\mathbb{R}^{(r+1)}}f(X)d^*X$ where $d^*X=\prod_{i=1}^{r+1}\frac{dx_i}{x_i}$.
We are interested in making the substitution $t=\prod_{i=1}^{r+1}x_i^{N_i}=||X||$ where $N_i=1$ if $i$ corresponds to a real embedding (equivalently $I\leq r_1$), and $N_i=2$ otherwise. We then define $c_i$ as the variable satisfying $x_i =t^{\frac{1}{n}} c_i$ so that $||C||=1$, so this transform just integrates around the shells of these $r+1$ dimensional "balls", kind of like transforming to polar.
Now the Jacobian must be calculated: the first $r$ rows/columns are simple: we see $\frac{\partial x_i}{\partial c_j}=\delta_{i,j}t^{\frac{1}{n}}$. Since we have the relation $||C||=1$, $C$ has only $r$ degrees of freedom so one $c_j$ (which we have arbitrarily taken to be $r+1$) is dependent on the other elements in the tuple. We have $c_{r+1}^{N_{r+1}}=\prod_{i=1}^{r}c_i^{-N_i}$ so $N_{r+1}c_{r+1}^{N_{r+1}-1}\frac{\partial c_{r+1}}{\partial c_j}=-\frac{N_j}{c_j}\prod_{i=1}^{r}c_i^{-N_i}$ and $\frac{\partial c_{r+1}}{\partial c_j}=-\frac{N_jc_{r+1}}{N_{r+1}c_j}$. Now we compute $\frac{\partial x_{r+1}}{\partial c_j}=t^{\frac{1}{n}}\frac{\partial c_r+1}{\partial c_j}=t^{\frac{1}{n}}\frac{N_jc_{r+1}}{N_{r+1}c_j}$. We also have $\frac{\partial x_{i}}{\partial t}=J_{I,r+1}=\frac{t^{\frac{1}{n}-1}}{n}c_i$. Now we have something close to an upper triangular matrix with a nonzero bottom row, we we can row reduce by adding each row above $i=r+1$ multiplied by $\frac{N_jc_{r+1}}{N_{r+1}c_j}$ (respectively) to the bottom row ($i=r+1$). This leaves us with a upper triangular matrix with a diagonal of $t^{\frac{1}{n}}$ in every entry besides the last, where we now have $J_{r+1,r+1}=\frac{\partial x_{r+1}}{\partial t}+\sum_{i=1}^{r}\frac{\partial x_{i}}{\partial t}=\frac{t^{\frac{1}{n}}c_{r+1}}{ntN_{r+1}}(\sum_{i=1}^{r}{N_i}+1)=\frac{t^{\frac{1}{n}}c_{r+1}}{tN_{r+1}}$. Finally we have $det(J)=\frac{t^{\frac{r+1}{n}}c_{r+1}}{tN_{r+1}}$, and $d^*X=\prod_{i=1}^{r+1}\frac{1}{t^{\frac{1}{n}}c_i}det(J)dCdt=\prod_{i=1}^{r}\frac{dc_i}{c_i}\frac{dt}{tN_{r+1}}$.
In the paper, this was almost the result--what was mentioned was the exact same without the $N_{r+1}$ in the denominator. I wanted to post this as a sanity check, I've redone the calculations several times and always arrive at this. Any help would be greatly appreciated.