Bombieri-Vinogradov for multiplicative functions, and beyond the x                         1/2                         -barrier

Andrew Granville; Xuancheng Shao

doi:10.1016/j.aim.2019.04.055

Bombieri-Vinogradov for multiplicative functions, and beyond the x ^1/2 -barrier

Andrew Granville
, Xuancheng Shao

Mathematics

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

9 Scopus citations

Abstract

Part-and-parcel of the study of “multiplicative number theory” is the study of the distribution of multiplicative functions in arithmetic progressions. Although appropriate analogies to the Bombieri-Vingradov Theorem have been proved for particular examples of multiplicative functions, there has not previously been headway on a general theory; seemingly none of the different proofs of the Bombieri-Vingradov Theorem for primes adapt well to this situation. In this article we find out why such a result has been so elusive, and discover what can be proved along these lines and develop some limitations. For a fixed residue class a we extend such averages out to moduli [Formula Presented].

Original language	English
Pages (from-to)	304-358
Number of pages	55
Journal	Advances in Mathematics
Volume	350
DOIs	https://doi.org/10.1016/j.aim.2019.04.055
State	Published - Jul 9 2019

Bibliographical note

Publisher Copyright:
© 2019 Elsevier Inc.

Funding

A.G. has received funding from the European Research Council grant agreement no 670239, and from NSERC Canada under the CRC program.X.S. was supported by a Glasstone Research Fellowship.

Funders	Funder number
Horizon 2020 Framework Programme	670239
Natural Sciences and Engineering Research Council of Canada
National Council for Eurasian and East European Research

Keywords

Bombieri-Vinogradov theorem
Multiplicative functions
Siegel zeroes

ASJC Scopus subject areas

General Mathematics

Access to Document

10.1016/j.aim.2019.04.055

Cite this

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title = " Bombieri-Vinogradov for multiplicative functions, and beyond the x 1/2 -barrier ",

abstract = "Part-and-parcel of the study of “multiplicative number theory” is the study of the distribution of multiplicative functions in arithmetic progressions. Although appropriate analogies to the Bombieri-Vingradov Theorem have been proved for particular examples of multiplicative functions, there has not previously been headway on a general theory; seemingly none of the different proofs of the Bombieri-Vingradov Theorem for primes adapt well to this situation. In this article we find out why such a result has been so elusive, and discover what can be proved along these lines and develop some limitations. For a fixed residue class a we extend such averages out to moduli [Formula Presented].",

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