I'm looking for applications of Fourier Transforms in number theory.
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$\begingroup$ Just applications of the Fourier transform itself or of more abstract harmonic analysis (e.g. over locally compact groups etc.)? There is a really huge body of work using abstract harmonic analysis in number theory. $\endgroup$– M.G.Commented Dec 2, 2017 at 1:15
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5$\begingroup$ Even ordinary Fourier analysis is ubiquitous in analytic number 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. ElkiesCommented Dec 2, 2017 at 1:26
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7$\begingroup$ You might take a look at: Montgomery, Hugh L. Ten 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 SilvermanCommented Dec 2, 2017 at 1:57
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$\begingroup$ Prime counting functions such as the fundamental prime counting function $\pi(x)$, Riemann's prime-power counting function $\Pi(x)$, and the second Chebyshev function $\psi(x)$ all have Fourier series representations. Please see math.stackexchange.com/q/2380164 for a fair amount of insight into the theory and value of Fourier series representations of non-periodic functions. $\endgroup$– Steven ClarkCommented Mar 2, 2018 at 21:43
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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 $\displaystyle J(e^u) e^{-\sigma u} = e^{-\sigma u}\sum_{p^k \le e^u} \frac{1}{k}=e^{-\sigma u}\sum_{k\ge 1} \frac{\pi(e^{u/k})}{k}$
