Beyond the Erdős discrepancy problem in function fields

Abstract

We characterize the limiting behavior of partial sums of multiplicative functions f : Fq[t] → S 1 . In contrast to the number field setting, the characterization depends crucially on whether the notion of discrepancy is defined using long intervals, short intervals, or lexicographic intervals.Concerning the notion of short interval discrepancy, we show that a completely multiplicative f : Fq[t] → {−1, +1} with q odd has bounded short interval sums if and only if f coincides with a "modified" Dirichlet character to a prime power modulus. This confirms the function field version of a conjecture over Z that such modified characters are extremal with respect to partial sums.Regarding the lexicographic discrepancy, we prove that the discrepancy of a completely multiplicative sequence is always infinite if we define it using a natural lexicographic ordering of Fq[t]. This answers a question of Liu and Wooley.Concerning the long sum discrepancy, it was observed by the Polymath 5 collaboration that the Erdős discrepancy problem admits infinitely many completely multiplicative counterexamples on Fq [t]. Nevertheless, we are able to classify the counterexamples if we restrict to the class of modified Dirichlet characters. In this setting, we determine the precise growth rate of the discrepancy, which is still unknown for the analogous problem over the integers.

Correlations of multiplicative functions in function fields

“…Our focus in this paper is on analogs of (1) and (2) over 𝔽 𝑞 [𝑡]. These results have applications, in particular, to the Erdős discrepancy problem over 𝔽 𝑞 [𝑡], which we study in our follow-up paper [24]. In the course of the proofs of our main results, we develop a substantial amount of pretentious number theory over 𝔽 𝑞 [𝑡].…”

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

“…In a fine work [17], Klurman provided an asymptotic formula for the sum (1) for two multiplicative functions 𝑓, g ∶ ℕ → 𝕌 with 𝔻(𝑓(𝑛), 𝑛 𝑖𝑡 1 𝜒(𝑛), ∞) < ∞ and 𝔻(g(𝑛), 𝑛 𝑖𝑡 2 𝜓(𝑛), ∞) < ∞ for some primitive Dirichlet characters 𝜒, 𝜓 and some real numbers 𝑡 1 , 𝑡 2 . As an application of this result, together with Tao's theorem (2), Klurman [17] proved Kátai conjecture (see [15]) that if 𝑓 ∶ ℕ → 𝕊 1 is completely multiplicative where 𝕊 1…”

“…The next two theorems show that global correlations are asymptotically products of local correlations. The first one is a function field analog of a result of Klurman [17].…”

“…In [18], a function field analog of Kátai conjecture has been established. The following theorem gives another proof using techniques from the proof of Kátai conjecture for integers (see [17]).…”