Where does the identity $p^{-ns}=s\int_{p^n}^\infty x^{-s-1}dx$ come from?

Question

In this paper, equation 45 (page 11) gives the identity

$$p^{-ns}=s\int_{p^n}^\infty x^{-s-1}dx$$

Can someone tell me where this comes from, and how it is derived?

Answer 1

This statement holds only for $s>0$.

For $s\ne0$ the anti-derivative of $x^{-s-1}$ is $-\frac{x^{-s}}s$. Applying the limits,$$\frac s{-s}[x^{-s}]_{p^n}^\infty=p^{-ns}$$since $\infty^{-s}=1/\infty^s=0$ for positive $s$.

Answer 2

Try to see it follows from identity 41,42,43 I see that it can be connected to the mellin transform of the function