In number theory, Terry Tao already mentioned Quadratic Reciprocity in his first comment, but there's also the reciprocity formula $$ s(b,c) + s(c,b) = \frac1{12}\left( \frac{b}{c} + \frac1{bc} + \frac{c}{b} \right) - \frac14 $$ for Dedekind sums, symmetrized further in Rademacher's formula $$ D(a,b;c) + D(b,c;a) + D(c,a;b) = \frac1{12} \frac{a^2+b^2+c^2}{abc} - \frac14. $$ [Here $D(a,b;c) = \sum_{n\,\bmod\,c} ((an/c)) ((bn/c))$, where $((\cdot))$ is the sawtooth function taking $x$ to $0$ if $x \in {\bf Z}$ and to $x - \lfloor x \rfloor - 1/2$ otherwise; and the Dedekind sum is the special case $s(b,c) = D(1,b;c)$.]