Can we expect that $ \int_{\mathbb T} \left| \sum_{n\in \mathbb N} e^{-in^{r}} e^{inx} \right| dx $ is finite for some $r>0$?

Question

Let $\mathbb T$ denotes the torus and $r>0.$

My Question is: Can we expect that $$ \int_{\mathbb T= [0, 2\pi)} \left| \sum_{n\in \mathbb N} e^{-in^{r}} e^{inx} \right| dx $$ is finite for some $r>0$? If so, how to justify or any known referneces (approach) to treat the problem?

Edit: $\mathbb T =\{z\in \mathbb C: |z|=1\}$ simply circle group.

Answer 1

Your expression as written is simply undefined. All terms in your sum have absolute value $1$, so the sum diverges.

It is possible, however, that some summability method for divergent series could be used to make sense of it.