Mobius function in short intervals for function fields

Abstract

Let mu(A) be the Mobius function defined in a polynomial ring F-q[T] with coefficients in the finite field F-q 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 Mobius 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.