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Harald,

My personal stance on this is that one should I like to try and avoid using Perron's formula in the "traditional" form. Instead, I like to see the Prime Number Theorem (say) as a statement about $\sum \Lambda(n) \phi(n)$, where $\phi$ is a $C^{\infty}_0$ cutoff function approximating the interval $[1,X]$. To relate this to $\zeta'/\zeta$, you need the Mellin inversion formula for $\phi$ on the vertical line $\Re s = \sigma$, and this really is precisely the same thing as the Fourier inversion formula for the function $e^{\sigma u}\phi(e^u)$. Since everything is a compactly supported smooth function, and in particular a Schwartz function, the analytic issues involved with inverting the Fourier transform are as mild as they can be.

My point of view on this is elaborated upon in in chapter 1 of this course http://www.dpmms.cam.ac.uk/~bjg23/ANT.html.

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Harald,

My personal stance on this is that one should try and avoid using Perron's formula in the "traditional" form. Instead, I like to see the Prime Number Theorem (say) as a statement about $\sum \Lambda(n) \phi(n)$, where $\phi$ is a $C^{\infty}_0$ cutoff function approximating the interval $[1,X]$. To relate this to $\zeta'/\zeta$, you need the Mellin inversion formula for $\phi$ on the vertical line $\Re s = \sigma$, and this really is precisely the same thing as the Fourier inversion formula for the function $e^{\sigma u}\phi(e^u)$. Since everything is a compactly supported smooth function, and in particular a Schwartz function, the analytic issues involved with inverting the Fourier transform are as mild as they can be.

My point of view on this is elaborated upon in in chapter 1 of this course http://www.dpmms.cam.ac.uk/~bjg23/ANT.html.