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arXiv · 1607.01814

Gowers norms of multiplicative functions in progressions on average

Xuancheng Shao
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Abstract

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

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Xuancheng Shao. 2017-01-31. Gowers norms of multiplicative functions in progressions on average. https://doi.org/10.2140/ant.2017.11.961

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