I feel that there is a good chance to apply certain integrals of $\ \exp(-(\sum_{k=1}^n x_k))\ $ over corners (see below) to the analytic number theory. I have obtained two formulas to start with but am asking about their joint generalization. If all this is well known then let me be in the known too (and I apologize).

A real sequence $\ \mathbf A:=(A_1\ \ldots\ A_n)\in(0;\infty)^n\ $ defines a corner

$$ \Delta(\mathbf A)\ :=\ \{ (x_1\ \ldots\ x_n)\in[0;\infty)^n\ : \ \sum_{k=1}^n\frac{x_k}{A_k}\ \le\ 1 \} $$

The special $n$-dimensional case is $\ \mathbf S\ :=\ (S\ \dots\ S),\ $ where $\ A_1=\ldots=A_n=S>0.\ $ I'll write $\ \Delta(n;\ S):=\Delta(\mathbf S)\ $ to show the dimension $n$ explicitly. Then,

**THEOREM 1**

$$ \int_{\Delta(n;\ S)} \exp\left(-\sum_{k=1}^n x_k\right)\cdot dx_1\ldots dx_n\,\ = \,\ 1 -\ \sum_{k=0}^{n-1} \frac{S^k}{k!} \cdot e^{-S} $$

Now let's consider the general but only 2-dimensional case, where $\ \mathbf A\ :=\ (A\ B),\ $ so that we can write $\ \Delta(A\ B):= \Delta(\mathbf A)\ $. Then,

**THEOREM 2**

$$ \int_{\Delta(A\ B)} \exp(-(x+y))\cdot dxdy\,\ = \,\ 1\ +\ \frac B{A-B}\cdot e^{-A}\ +\ \frac A{B-A}\cdot e^{-B} $$

*See the comment below about the case* $\ A=B>0$.

Q U E S T I O N

I'd welcome the n-dimensional formula for the integral over the n-dimensional corner, especially in the spirit of Theorem 2 above.

A comment:Singularity (removable;)

Theorem 2 doesn't look kosher when $\ A=B>0.\ $ But it is a $2$-dimensional generalization of Theorem 1. To see it, let $\ D:=B-A.\ $ Then

$$ \int_{\Delta(A\ B)} \exp(-(x+y))\cdot dxdy\,\ = \,\ 1\ - \left(1+\frac{1-e^{-D}}D\cdot A\right)\cdot e^{-A} $$

Thus, for $\ D\rightarrow 0,\ $ the above expression approaches the 2-dimensional case of Theorem 1.