Fourier transform of the von Mangoldt function? Wikipedia states under the entry for the von Mangoldt function:
The Fourier transform of the von Mangoldt function gives a spectrum with spikes at ordinates equal to imaginary part of the Riemann zeta function zeros.
(I believe "ordinates" should be changed to "abscissas".)
First, what does taking the Fourier transform of the von Mangoldt function mean?
Second, if meaningful, is it true?   
Third, if true, how might this be related to the sum of the exponentials $e^{iImg(z_n)x}$ over the non-trivial zeros $z_n$ above and below the real axis, assuming the RH is true?
(There is some history behind this statement in an older MO-Q, but the analysis there is not clear to me either.) 
 A: Start with the explicit formula
$$\sum_{n \le x}\Lambda(n) =\frac1{2i\pi} \int_{2-i\infty}^{2+i\infty} \frac{-\zeta'(s)}{\zeta(s)}\frac{x^s}s ds=1_{x > 1}\sum Res(\frac{-\zeta'(s)}{\zeta(s)}\frac{x^s}s)$$ $$=1_{x > 1}( x - \sum_\rho \frac{x^\rho}{\rho} - \frac12 \log 2\pi - \sum_{k=1}^\infty \frac{x^{-2k}}{-2k})$$
Since $\sum_\rho \frac1{|\rho|^2}<\infty$  the RHS converges in $L^1_{loc}$ thus we can differentiate both sides in the sense of distributions, the RHS being continuous at $1$ we get 
$$\sum_n \Lambda(n) \delta(x-n) =1_{x > 1} - 1_{x > 1}\sum_\rho x^{\rho-1} +\frac{d}{dx} 1_{x > 1}\log(1-x^{-2})$$
If the RH is true,  letting $x =e^u$ and multiplying both side by $e^{u/2}$, since $e^{u/2}\delta(e^u-n) = \frac{\delta(u-\log n)}{n^{1/2}}$ 
$$\sum_n  \frac{\Lambda(n)}{n^{1/2}}\delta(u-\log n) =1_{u > 0}e^{u/2} - 1_{u > 0}\sum_t e^{itu}  +e^{-u/2}\frac{d}{du}1_{u > 0} \log(1-e^{-2u}))$$
Making both side even 
$$\sum_n \frac{\Lambda(n)}{n^{1/2}} (\delta(u-\log n)+\delta(u+\log n))=e^{|u|/2} - \sum_t e^{itu}  -e^{-|u|/2}(\frac{d}{d|u|} \log(1-e^{-2|u|})$$

Which means that we have the Fourier transform in the sense of distributions $$\mathcal{F}^{-1}\left[2 \pi \sum_t \delta(\omega-t)\right] = e^{|u|/2}-e^{-|u|/2}(\frac{d}{d|u|} \log(1-e^{-2|u|})-\sum_n \frac{\Lambda(n)}{n^{1/2}} (\delta(u-\log n)+\delta(u+\log n)) $$

If the RH is not true then those things are true only in the sense of analytic functionals, for example they hold when using Gaussians as test functions.
