The explicit formula for $\zeta(s)$: $$ \psi(x)=x-\sum_{|\operatorname{Im}\rho|<T}\frac{x^\rho}{\rho}-\log(2\pi)-\log\left(1-\frac{1}{x^2}\right)+O\left(\frac{x\log^2T}{T}\right), $$ where $\psi(x)=\sum_{p^k<x}\log p$, $x>1$, $\rho$ is a nontrivial zero of $\zeta(s)$, and the sum over $\rho$ is taken with multiplicities.

$\textbf{My question}:$ Is there any literature on the behavior of the equality when we take formal derivatives on both sides? I am looking for asymptotic results after taking the derivative on both sides.

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    $\begingroup$ I'm not quite sure what a formal derivative of the left hand side even means. The left hand side is discrete and locally constant, except at integers. What do you want the derivative of the left hand side to mean? I suppose one possibility would be to differentiate the Mellin representation of the left hand side, which would exist. But is that what you want? $\endgroup$ – davidlowryduda Apr 24 '17 at 2:53
  • $\begingroup$ @mixedmath: I don't have any specific representation in mind. Any sort of interpretation for the derivatives will work for me. Can you please point me to some references regarding the differentiation of Mellin representation that you have stated? $\endgroup$ – James Taylor Apr 24 '17 at 4:38
  • $\begingroup$ The derivative of $\psi(x)$ is a distribution, a sum of Dirac deltas, and the derivative of the RHS converges to this distribution in the sense of distributions. You should work on $\sum_{k=-\infty}^\infty \delta(x-k) = \sum_{n=-\infty}^\infty e^{2i \pi n x}$ which is the fundamental theorem of Fourier series encoded in term of distributions. $\endgroup$ – reuns Apr 24 '17 at 18:31

(Expanding on my comment)

The classical way of recognizing this explicit formula is to perform a careful analysis of the integral $$ \psi(X) = \frac{1}{2\pi i} \int_{5 - i\infty}^{5 + i\infty} \left( - \frac{\zeta'(s)}{\zeta(s)} \right) X^s \frac{ds}{s}. $$ The convergence of this integral is a bit delicate, but it may be possible to define $\psi'(X)$ by defining it to be the value of the $X$-derivative of the integral.

A simpler problem would be to try to understand the integral $$ \frac{d}{dX} \frac{1}{2\pi i} \int_{5 - i\infty}^{5 + i\infty} X^s \frac{ds}{s}. $$ This behaves very well away from integer values of $X$, but at integer values this requires some interpretation.


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