Bounds for truncated $L$-series over short intervals

Let $\chi$ be a non-principal Dirichlet character. Are there any good non-trivial bounds for short sums of the form $$\sum_{x < n \leq x + N} \chi(n)n^{i t}$$ as both $x \geq 1$ and $t \in \mathbb{R}$ grow large and $N \ll x$? In the case when $|t|$ is small relative to $x$, summation by parts together with bounds for sums of $\chi(n)$ work quite well. But what in the case when $t$ is also large, say $t \gg x$? Thanks.

• I think that for large $x$ and $\lvert t\rvert$ you can't do better (at least not significantly better) than the trivial $\lvert \dotsc\rvert \leqslant N$ (times approximately $\frac{\varphi(k)}{k}$ where $k$ is the period of $\chi$). For large $x$, there's a good chance that $\{\log n : x < n \leqslant x + N\}$ is linearly independent over $\mathbb{Q}$, and then you can find $t$ such that $\chi(n)n^{it}$ is close to $1$ (or $= 0$) for all these $n$. – Daniel Fischer Aug 26 '18 at 18:15