Is there a known expression for the (distributional) Fourier transform of the Riemann zeta function, taken along the critical line?
I'd love to say that it's a weighted sum of delta distributions, logarithmically spaced and decreasing in amplitude, as in
$\sum_n \frac{\delta(\omega-\log(n))}{n^{1/2}}$
but this fails to be a tempered distribution, and fails in general when the exponent in the denominator is less than 1.