When proving that every natural number is a sum of four squares, one puts the number of possibilities (counted in the right way!) $r_4(n)$ into a series $\sum r_4(n)q^n$ which turns out to be a modular form (a "familiy of coefficients") of some level and can therefore be written as a sum of suitable Eisenstein series identifying the $r_4(n)$ with simpler terms.