Möbius function in short intervals for function fields

Let μ(A) be the Möbius function defined in a polynomial ring Fq[T] with coefficients in the finite field Fq of q elements (q is odd). In this paper, we present a function field version of partial progress toward a conjecture of Good and Churchhouse. We calculate the mean and the large q limit of the variance of partial sums of the Möbius function on short intervals. Our calculation closely follows the framework of a recent work of Keating and Rudnick, where they consider the distribution of the von Mangoldt function in function fields.