I'm trying to understand Riemann's "Über die Anzahl der Primzahlen unter einer gegebenen Größe" ("On the Number of Prime Numbers less than a Given Quantity" in english). Here is the original work in German and here the english translation.
I'm stuck right on page 1 already at this identity:
$ \int_0^\infty e^{-nx}x^{s-1}dx = \frac{\Pi(s-1)}{n^s} $
I understand that the left side (without the "n"):
$ \int_0^\infty e^{-x}x^{s-1}dx $
is the Gamma function and the right hand side (without the denominator):
$ \Pi(s-1) $
is the Pi function - so the Gamma function when offset to coincide with the factorial.
For the gamma function holds:
$ \Gamma(n) = (n-1)! $
and for the Pi function holds:
$ \Pi(n) = n! $
and thus
$ \Pi(n-1) = (n-1)! $
which means that:
$ \Gamma(n) = \Pi(n-1) $
Which means I can write this also down like that:
$ \int_0^\infty e^{-x}x^{s-1}dx = \Pi(s-1) $
which is close to the original identity from above.
But where does the "n" come from on both sides?
I don't understand how I get from
$ \int_0^\infty e^{-x}x^{s-1}dx = \Pi(s-1) $
to
$ \int_0^\infty e^{-nx}x^{s-1}dx = \frac{\Pi(s-1)}{n^s} $
And what does the "n" mean in this context? It's neither an argument to the function nor is it the variable which we are integrating with respect to.