What I like(d) most is defining an analytic function that describes some number theoretic phenomena. One thing I remember is from Winfried Kohnen's postech lecture http://www.mathi.uni-heidelberg.de/~winfried/siegel2.pdf , see pages 1-3 for more details.
He starts with the standard inner product on $\mathbb{R}^m$ viewed as a quadratic form $$Q(x):=x^t x.$$
We are interested in the number $r_Q(t)$ of tuples of squares of inetegers that add up to a natural number $t$, i.e.

$$ r_Q(t):= \# \left\{ g \in \mathbb{Z}^m : Q(g)=(g_1)^2+ \dots + (g_4)^2=t \right\} .$$

They can be computed via this power series

$$ \theta_Q(z) = 1+ \sum_{t\geq 1} r_Q(t)\ \exp(2\pi i tz) $$

that is in fact $\theta_Q$ is a modular form of weight 2 w.r.t. $\Gamma_0$. Therefore (ok here is some kind of black box for the students), its Fourier coefficients can be given by
$$r_Q(t)= 8 \left( \sigma_1(t)-4\cdot \sigma_1\left(\frac{t}{4}\right) \right)$$

where $\sigma_k(t)$ denotes the divisor function $$\sigma_k(t):=\sum_{d|t} d^k.$$

While writing this I was wondering whether the prime number theorem and elegant proofs of the fundamental theorem of algebra are too well known.

p.s. sorry for messing up the formulas again.