Gowers norms of multiplicative functions in progressions on average

Xuancheng Shao

doi:10.2140/ant.2017.11.961

Gowers norms of multiplicative functions in progressions on average

Xuancheng Shao

Mathematics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

Let μ be the Möbius function and let k ≥ 1. We prove that the Gowers U^k-norm of μ restricted to progressions {n ≤ X : n ≡ a_q (mod q)} is o(1) on average over q ≤ X^1/2-^σ for any σ > 0, where a_q (mod 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.

Original language	English
Pages (from-to)	961-982
Number of pages	22
Journal	Algebra and Number Theory
Volume	11
Issue number	4
DOIs	https://doi.org/10.2140/ant.2017.11.961
State	Published - 2017

Bibliographical note

Publisher Copyright:
© 2017 Mathematical Sciences Publishers.

Keywords

Bombieri-Vinogradov theorem
Gowers norms
Multiplicative functions

ASJC Scopus subject areas

Algebra and Number Theory

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10.2140/ant.2017.11.961

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abstract = "Let μ be the M{\"o}bius function and let k ≥ 1. We prove that the Gowers Uk-norm of μ restricted to progressions {n ≤ X : n ≡ aq (mod q)} is o(1) on average over q ≤ X1/2-σ for any σ > 0, where aq (mod q) is an arbitrary residue class with (aq, q) = 1. This generalizes the Bombieri-Vinogradov inequality for μ, which corresponds to the special case k = 1.",

keywords = "Bombieri-Vinogradov theorem, Gowers norms, Multiplicative functions",

author = "Xuancheng Shao",

note = "Publisher Copyright: {\textcopyright} 2017 Mathematical Sciences Publishers.",

year = "2017",

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language = "English",

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pages = "961--982",

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T1 - Gowers norms of multiplicative functions in progressions on average

AU - Shao, Xuancheng

PY - 2017

Y1 - 2017

N2 - Let μ be the Möbius function and let k ≥ 1. We prove that the Gowers Uk-norm of μ restricted to progressions {n ≤ X : n ≡ aq (mod q)} is o(1) on average over q ≤ X1/2-σ for any σ > 0, where aq (mod q) is an arbitrary residue class with (aq, q) = 1. This generalizes the Bombieri-Vinogradov inequality for μ, which corresponds to the special case k = 1.

AB - Let μ be the Möbius function and let k ≥ 1. We prove that the Gowers Uk-norm of μ restricted to progressions {n ≤ X : n ≡ aq (mod q)} is o(1) on average over q ≤ X1/2-σ for any σ > 0, where aq (mod q) is an arbitrary residue class with (aq, q) = 1. This generalizes the Bombieri-Vinogradov inequality for μ, which corresponds to the special case k = 1.

KW - Bombieri-Vinogradov theorem

KW - Gowers norms

KW - Multiplicative functions

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VL - 11

SP - 961

EP - 982

JO - Algebra and Number Theory

JF - Algebra and Number Theory

IS - 4

ER -

Gowers norms of multiplicative functions in progressions on average

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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