# Sum over powers of roots of unity gives only partial answer

Mathematica seems to be unable to handle sums over roots of unity. For example, Simplify[Sum[Exp[2 Pi I a b/m], {a, m}], {b, m} \[Element] Integers] yields 0 instead of a piecewise function like:$$\begin{cases} 0 & gcd(b,m)=1\ m & \text{otherwise} \end{cases} \$$ Has anybody managed to obtain the correct answer?