  We study the $q$-analogue of the average of Montgomery's function $F(\alpha,
T)$ over bounded intervals. Assuming the Generalized Riemann Hypothesis for
Dirichlet $L$-functions, we obtain upper and lower bounds for this average over
an interval that are quite close to the pointwise conjectured value of 1. To
compute our bounds, we extend a Fourier analysis approach by Carneiro, Chandee,
Chirre, and Milinovich, and apply computational methods of non-smooth
programming.
