An integral involving two variables and the floor function

Question

Let $N$ be some fixed positive integer. I have the following function $$ g(z) = z \int_1^N [t] e^{2 \pi i t z} \ dt. $$ How would one compute $$ \int_0^1 g(z) \ dz ? $$ Thanks!

Answer 1

My first instinct: break $g(z)$ into parts, so $$g(z) = z \sum_{k=1}^{N-1} \int_{t=k}^{k+1} k e^{2 \pi i t z} \, dt = z \sum_{k=1}^{N-1} k \biggl[ \frac{e^{2 \pi i t z}}{2 \pi i z} \biggr]_{t=k}^{k+1}.$$ Then see if you can simplify this sum, and then integrate it with respect to $z$.

Answer 2

First change the order of integration and evaluate the inner integral which is with respect to $z$ using integration by parts, then advance to evaluate the integral involving the floor function. See related techniques.