In Riemann's Paper On the Number of Prime Numbers less than a Given Quantity he states the Euler Product
$\prod\frac{1}{1-\frac{1}{p^s}}=\sum\frac{1}{n^s}$
"Where if one substitutes for $p$ all prime numbers, and for n all whole numbers. The function of the complex variable a which is represented by these two expressions, wherever they converge, I denote $\zeta(s)$. Both expressions converge only when the real part of $s$ is greater than 1." This is from the paper.
He then uses the equation
$$\tag{1}\int_0^\infty e^{-nx}x^{s-1}dx=\frac{\prod(s-1)}{n^{s}} $$
$$\tag{2}{\prod(s-1)}\zeta(s)=\int_0^\infty \frac{x^{s-1}}{e^{x}-1}dx$$
My question is how goes from equation 1 and 2 and was wondering if anyone could explain it.
Thanks me
You sum both sides of the equality $(1)$:\begin{align}\sum_{n=1}^\infty\int_0^\infty e^{-nx}x^{s-1}\,\mathrm dx&=\int_0^\infty\sum_{n=1}^\infty e^{-nx}x^{s-1}\,\mathrm dx\\&=\int_0^\infty\frac{e^{-x}}{1-e^{-x}}x^{s-1}\,\mathrm dx\\&=\int_0^\infty\,\frac{x^{s-1}}{e^x-1}\mathrm dx\end{align}and$$\sum_{n=1}^\infty\frac{\Pi(s-1)}{n^s}=\Pi(s-1)\zeta(s).$$