What is the Fourier transform of Riemann Zeta function? All:
Is there an explicit form of Fourier Transform of Riemann Zeta function ?
Also, is there an discrete Fourier Transform (DFT) of Riemann Zeta function ?
I remembered I had seen something like this, but I could not remember.
Thank you in advance for your help.
 A: Riemann zeta, $\zeta(s)$ is coming from prime numbers and these have multiplicative structure. Therefore it is not useful to connect them using ordinary Fourier transform to anything. It is much better to use Mellin transform which is connected to Fourier transform almost directly, we just use the logarithmic scaling more or less.
So if you define Mellin as
$$\left\{\mathcal{M}f\right\}(s) = \phi(s)=\int_0^\infty x^{s-1} f(x) \, dx$$
$$\left\{\mathcal{M}^{-1}\phi\right\}(x) = f(x)=\frac{1}{2 \pi i} \int_{c-i \infty}^{c+i \infty} x^{-s} \phi(s)\, ds$$
then you have
$$ f(x) = \frac{1}{e^x-1} $$
$$\left \{ \mathcal Mf \right \}(s)=\Gamma(s)\zeta(s)$$
Regarding discrete, it is not clear over what you would want to do the transformation so I would pass to related Hurwitz zeta $\zeta(s,a)$ where the two are related over $\zeta(s,1)=\zeta(s)$.
We know that these two bellow are a discrete Fourier pair:
$$\zeta(\nu ,a)=\sum_{k=0}^\infty \frac {1}{(k+a)^\nu },\quad 0<a\le 1,\operatorname {Re} \nu >1$$
$$\chi_\nu (z)=\sum_{k=0}^\infty \frac {z^{2k+1}}{(2k+1)^\nu },\quad |z|\le 1,\operatorname {Re} \nu >1 \text { with } \nu =2,3,4,\dotsc$$
