Theorem 18.13 of Montgomery and Vaughan's multiplicative number theory book says for non-principal characters (ignoring $\log $'s) $$\int _{-\delta }^{\delta }|\psi _\chi (\beta )|^2d\beta \ll \delta x\hspace {15mm}\psi _\chi (\beta ):=\sum _{n<x}\Lambda (n)\chi (n)e(n\beta ).$$
I would like to know if this stays true if we replace the interval $(-\delta ,\delta )$ on the LHS with another interval $(x_0-\delta ,x_0+\delta )$ of the same length.
It doesn't seem to me that $\beta \in (-\delta ,\delta )$ in $\psi _\chi (\beta )$ seems to have nothing particularly to do with $\Lambda (n)\chi (n)$, so I thought there's nothing special about this interval and it should hold with a shifted interval too. However this thinking is probably wrong because the proof doesn't seem to modify easily for a shift. But maybe I'm missing something.
