Gowers norms of multiplicative functions in progressions on average

TL;DR

This paper demonstrates that the Gowers norms of the Möbius function, when restricted to arithmetic progressions, tend to zero on average over moduli up to a certain size, extending classical distribution results.

Contribution

It generalizes the Bombieri-Vinogradov inequality for the Möbius function to higher Gowers norms, providing new insights into its pseudorandomness in arithmetic progressions.

Findings

Gowers norms of μ are o(1) on average over q ≤ X^{1/2 - σ}

Extension of Bombieri-Vinogradov inequality to higher Gowers norms

Results hold uniformly over residue classes coprime to q

Abstract

Let $\mu$ be the M\"{o}bius function and let $k \geq 1$. We prove that the Gowers $U^k$-norm of $\mu$ restricted to progressions $\{n \leq X: n\equiv a_q\pmod{q}\}$ is $o(1)$ on average over $q\leq X^{1/2-\sigma}$ for any $\sigma > 0$, where $a_q\pmod{q}$ is an arbitrary residue class with $(a_q,q) = 1$. This generalizes the Bombieri-Vinogradov inequality for $\mu$, which corresponds to the special case $k=1$.