Firstly, please, do not regard me as any kind of authority in the topics of fourier analysis or number theory (I'm 17 y.o. without any formal degree in mathematics), but let me show you an example of usage of Fourier analysis in number theory which, in my opinion, is simply beautiful.
A very anticipated conjecture is the one of Goldbach, stating that for any even number $N \geqslant 4$ the equation
$$N = p_1 + p_2$$
has a solution with $p_1,p_2$ being prime numbers. A weaker version, known as the ternary Goldbach conjecture, states that for any odd number $N \geqslant 5$ the equation
$$N = p_1 + p_2 + p_3$$
has a solution with $p_1,p_2,p_3$ being primes. This result has been proven by Vinogradov for all sufficiently large $N$ in a very specific way. Vinogradov lets
$$S(\alpha) = \sum_{k \leqslant N}\Lambda(k)e(k\alpha)$$
where $\Lambda$ is von Mangoldt's function and $e(x) = e^{2 \pi i x}$. Then
$$S(\alpha)^3 = \sum_{n \leqslant 3N}\left(\sum_{k_1 + k_2 + k_3 = n; k_1,k_2,k_2\leqslant N}\Lambda(k_1)\Lambda(k_2)\Lambda(k_3)\right)e(n\alpha).$$
Denoting the coefficients by $r(n,N)$, we have
$$r(n,N) = \int_{0}^{1}S(\alpha)e(-n\alpha) d\alpha.$$
From this point the proof gets rather technical, but one can show that
$$r(n,N) = \frac{1}{2}G(n)n^2 + O\left(\frac{n^2}{(\log n)^A}\right)$$
which has a direct impact on the number of solutions $(p_1,p_2,p_3)$ to the equation in conjecture.
I thinks this already shows well-enough how some of the Fourier analysis methods can be used in number theory. A more recent example might be this paper by James Maynard, which shows that there are infinitely many prime numbers without one of the digits in their decimal representation. Here the proof is based on the Fourier transform of the set
$$ \mathcal{A}_1 = \left\{ \sum_{0 \leqslant i \leqslant k}n_{i}10^{i} \colon n_{i} \in \{0,\ldots,9\} \setminus \{a_0\}, k \geqslant 0 \right\}$$
for a given digit $a_{0} \in \{0,\ldots,9\}$.
I hope this helps :)
