Correlations of multiplicative functions in function fields

Abstract

We develop an approach to study character sums, weighted by a multiplicative function f : Fq[t] → S 1 , of the formwhere χ is a Dirichlet character and ξ is a short interval character over Fq [t]. We then deduce versions of the Matomäki-Radziwi l l theorem and Tao's two-point logarithmic Elliott conjecture over function fields Fq[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 Möbius function for various values of q.Compared with the integer setting, we encounter some different phenomena, specifically a low characteristic issue in the case that q is a power of 2, as well as the need for a wider class of "pretentious" functions called Hayes characters.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 integer case.In a companion paper, we will use these results to characterize the limiting behavior of partial sums of multiplicative functions in function fields and in particular to solve the "corrected" form of the Erdős discrepancy problem over Fq[t]. 1 A function f (G) is called a factorization function if it only depends on the values of deg(P ) and vP (G), where P runs through the irreducible divisors of G, and vP (G) denotes the largest integer k with P k | G.

Beyond the Erdős discrepancy problem in function fields

“…In the non-pretentious case, the main ingredient that we need is a function field version of Tao's result on two-point logarithmic correlations of multiplicative functions. This was established by the authors in [11].…”

“…Theorem 2.2 (Two-point logarithmic Elliott conjecture in function fields, [11]). Let B ∈ F q [t]\{0} be fixed.…”

“…To prove the second claim where f : M → {−1, +1}, we take the conclusion (11) and apply the triangle inequality to deduce that…”

An induction principle for the Bombieri-Vinogradov theorem over $\mathbb{F}_q[t]$ and a variant of the Titchmarsh divisor problem

“…A version of this has recently been given by Klurman, Mangerel and Teräväinen ([28, Lemma 4.2]). As we shall see in Section 3 (see the remark following Theorem 3.4), the estimate in [28] does not suffice for us and we require an upper bound which is better on average over Dirichlet characters χ. We proceed to prove this in Section 3 (see Theorem 3.5).…”