Correlations of multiplicative functions in function fields

Abstract

We develop an approach to study character sums, weighted by a multiplicative function f0pt:Fq[t]→S1$f\colon \mathbb {F}_q[t]\rightarrow S^1$, of the form ∑degfalse(Gfalse)=NG0.33emmonicffalse(Gfalse)χfalse(Gfalse)ξfalse(Gfalse),$$\begin{align*}\hskip7pc \sum _{\substack{\textnormal {deg}(G) = N \\ G \text{ monic}}}f(G)\chi (G)\xi (G), \end{align*}$$where χ is a Dirichlet character and ξ is a short interval character over Fq[t]$\mathbb {F}_q[t]$. We then deduce versions of the Matomäki–Radziwiłł theorem and Tao's two‐point logarithmic Elliott conjecture over function fields Fq[t]$\mathbb {F}_q[t]$, 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 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 Fq[t]$\mathbb {F}_q[t]$.

Correlation of multiplicative functions over Fq[x]$\mathbb {F}_q[x]$: A pretentious approach

“…Following Klurman [16], we define the "distance" between two multiplicative functions 𝜓 1 , 𝜓 2 ∶  → 𝕌 by…”

“…Recently, Klurman et al. [18] proved the following two‐point logarithmically averaged Chowla and Elliott conjecture over function fields: Let $$\psi _1,\psi _2:\mathcal {M}\rightarrow \mathbb {U}$$ be multiplicative functions. Assume that ψ 1 satisfies the nonpretentiousness assumption $$$$\begin{equation*} \min _{\theta \in [0, 1]}\mathbb {D}{\left(\psi _1(P), \chi (P)\xi (P)e^{2\pi i \theta \deg (P)}; N \right)}\rightarrow \infty , \end{equation*}$$$$as $$N\rightarrow \infty$$, for all Dirichlet character $$\chi \pmod M$$ with $$M\in \mathcal {M}_{\leqslant W}$$ for every fixed $$W \geqslant 1$$ and all short interval characters ξ of length $$\leqslant n$$.…”

“…Also, very recently, Klurman et al. [19] studied the Erdös discrepancy problem over function fields.…”

“…Tao's work on logarithmically averaged Chowla–Elliott conjecture [30] for integers shows that if a multiplicative function $$f:\mathbb {N}\rightarrow \mathbb {U}$$ does not pretend to be a twisted Dirichlet character $$\chi (n)n^{it}$$, then f has small autocorrelation, namely, $$$$\begin{equation*} \frac{1}{\log x}\sum _{n\leqslant x}\frac{f(n)f(n+h)}{n}=o(1). \end{equation*}$$$$The situation is markedly different in case of function fields as demonstrated in [18]. Their proof of logarithmically averaged Chowla and Elliott conjecture for function fields showed that obstruction to small autocorrelation comes from twisted Hayes characters that are the product of a Dirichlet character and a short interval character (details are in the next section).…”

“…In the large degree limit, Klurman [16] derived analogs of Wirsing, Halász, and Hall's theorem on 𝔽 𝑞 [𝑥]. A recent groundbreaking result of Sawin and Shusterman [28] established the Chowla conjecture for function fields stated in the form…”