Improved quadratic Gowers uniformity for the M\"obius function

TL;DR

This paper improves bounds on the Gowers uniformity norms of the Möbius and von Mangoldt functions, leading to more precise asymptotic formulas for certain prime configurations, using advanced quadratic Fourier analysis techniques.

Contribution

It provides the first quasi-polynomial bounds in quadratic Fourier analysis over inite cyclic groups, improving previous bounds and applying new inverse theorems.

Findings

Bounds on unctions' Gowers norms are significantly improved.

Asymptotic formula for 4-term arithmetic progressions with primes is established.

First application of quasi-polynomial bounds in quadratic Fourier analysis over inite cyclic groups.

Abstract

We demonstrate that $$\|\mu\|_{U^3([N])} \ll_{A}^{\text{ineff}} \log^{-A}(N)$$ $$\|\Lambda - \Lambda_Q\|_{U^3([N])} \ll_{A}^{\text{ineff}} \log^{-A}(N)$$ for any $A > 0$ where $\Lambda_Q$ is an approximant to the von Mangoldt function and will be defined below, improving upon a bound of Tao-Ter\"av\"ainen (2021). As a consequence, among other things, we have the following: $$\mathbb{E}_{x, y \in [N], x + 3y \in [N]} \Lambda(x)\Lambda(x + y)\Lambda(x + 2y)\Lambda(x + 3y) = \mathfrak{S} + O_A(\log^{-A}(N))$$ where $\mathfrak{S}$ is the singular series for the configuration $(x, x + y, x + 2y, x + 3y)$. In fact, we show that $$\|\mu - \mu_{Siegel}\|_{U^3([N])} \ll \exp(-O(\log^{1/C}(N)))$$ $$\|\Lambda - \Lambda_{Siegel}\|_{U^3([N])} \ll \exp(-O(\log^{1/C}(N)))$$ where $\mu_{Siegel}$ and $\Lambda_{Siegel}$ are approximants of $\mu$, and $\Lambda$, respectively, representing the Siegel zero contribution of $\mu$ and are defined in the above article. To do so, we use an improvement of the $U^3$ inverse theorem due to Sanders and we follow the approach of Green and Tao (2007), opting to use the ``old-fashioned" approach to equidistribution on two-step nilmanifolds which was also considered by Green and Tao (2017), and by Gowers and Wolf (2010). To the author's knowledge, this is the first time that quadratic Fourier analysis over $\mathbb{Z}/N\mathbb{Z}$ has achieved quasi-polynomial type bounds in applications.