We study the
q
q
-analogue of the average of Montgomery’s function
F
(
α
,
T
)
F(\alpha ,\, T)
over bounded intervals. Assuming the Generalized Riemann Hypothesis for Dirichlet
L
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
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.