I'm looking for applications of Fourier Transforms in number theory.

Do you know the Riemann zeta function ?

$\displaystyle\frac{\log \zeta(\sigma+2i\pi \xi)}{\sigma+2i\pi \xi}$ is the Fourier transform of the prime counting function $\displaystyle J(e^u) e^{-\sigma u} = e^{-\sigma u}\sum_{p^k \le e^u} \frac{1}{k}=e^{-\sigma u}\sum_{k\ge 1} \frac{\pi(e^{u/k})}{k}$

analyticnumber theory, including the analytic formulas for class numbers of quadratic number fields. For numerical methods, see Odlyzko et al on computing many zeros of the Riemann zeta function and analogues. $\endgroup$ – Noam D. Elkies Dec 2 '17 at 1:26Ten lectures on the interface between analytic number theory and harmonic analysis. CBMS Regional Conference Series in Mathematics, 84. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1994. xiv+220 pp. $\endgroup$ – Joe Silverman Dec 2 '17 at 1:57