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Yes, this is a well-studied problem, see under character sums in the literature. For example, the Pólya-Vinogradov inequality says that the sum is $\ll\sqrt{d}\log d$, while for $d$ square-free, Burgess's bound implies that the sum is $\ll_{r,\epsilon} d^{\frac{r+1}{4r^2}+\epsilon}X^{1-\frac{1}{r}}$ for any $\epsilon>0$ and any integer $r\geq 1$. The square-free assumption can be removed at the cost of slightly weakening the bound, by comparing the character for $d$ with the character for the square-free kernel of $d$. The quoted theorems are Theorems 12.5 and 12.6 in Iwaniec-Kowalski: Analytic number theory.