Correlations of multiplicative functions in function fields

Abstract

We develop an approach to study character sums, weighted by a multiplicative function f : F-q [t] -> S-1, of the formSigma(deg(G)=N G monic) f(G)chi 9G)xi(G),where chi is a Dirichlet character and xi is a short interval character over F-q[t]. We then deduce versions of the Matomaki-Radziwill theorem and Tao's two-point logarithmic Elliott conjecture over function fields F-q[t], where q is fixed. The former of these improves on work of Gorodetsky, and the latter extends the work of Sawin-Shusterman on correlations of the Mobius function for various values of q. Compared with the integer setting, we encounter a different phenomenon, specifically a low characteristic issue in the case that q is a power of 2. As an application of our results, we give a short proof of the function field version of a conjecture of Katai on classifying multiplicative functions with small increments, with the classification obtained and the proof being different from the existing one in the integer case. In a companion paper, we use these results to characterize the limiting behavior of partial sums of multiplicative functions in function fields and in particular to solve a "corrected" form of the Erdos discrepancy problem over If F-q[t].