Correlations of multiplicative functions in function fields

Abstract

Abstract We develop an approach to study character sums, weighted by a multiplicative function , of the form where χ is a Dirichlet character and ξ is a short interval character over . We then deduce versions of the Matomäki–Radziwiłł theorem and Tao's two‐point logarithmic Elliott conjecture over function fields , 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 Möbius 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 Kátai 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 Erdős discrepancy problem over .

Locations

• arXiv (Cornell University) - View - PDF
• Durham Research Online (Durham University) - View - PDF
• DataCite API - View
• Mathematika - View - PDF