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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?

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  • $\begingroup$ For $s\ne0$ the anti-derivative of $x^{-s-1}$ is $-\frac{x^{-s}}s$. $\endgroup$ – Shubham Johri Jan 7 at 15:41
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    $\begingroup$ Thanks @Shubham Johri. Do you want to turn this into an answer so I can tick it? $\endgroup$ – Richard Burke-Ward Jan 7 at 15:43
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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 $s$.

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

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