WHEN DOES THE BOMBIERI–VINOGRADOV THEOREM HOLD FOR A GIVEN MULTIPLICATIVE FUNCTION?

Andrew Granville,Xuancheng Shao

doi:10.1017/fms.2018.14

Abstract

Let $f$ and $g$ be 1-bounded multiplicative functions for which $f\ast g=1_{.=1}$. The Bombieri–Vinogradov theorem holds for both $f$ and $g$ if and only if the Siegel–Walfisz criterion holds for both $f$ and $g$, and the Bombieri–Vinogradov theorem holds for $f$ restricted to the primes.

Highlights

In many examples it is difficult to obtain a strong bound on ∆( f, x; q, a) for arithmetic progressions modulo a particular q but one can perhaps do better on ‘average’
In this article we focus on 1-bounded multiplicative functions f ; that is, those f for which | f (n)| 1 for all n 1
The Bombieri–Vinogradov hypothesis holds for f · 1P, and the Siegel–Walfisz criterion holds for both f and g

Summary

Introduction

Many of the proofs of the Bombieri–Vinogradov theorem (for example, those going through Vaughan’s identity) relate the distribution of 1P in arithmetic progressions to the distribution of μ(.) in arithmetic progressions; here μ denotes the Mobius function, the convolution inverse of the multiplicative function 1 This is the prototypical example of the phenomenon we discuss in this article. The Bombieri–Vinogradov hypothesis holds for f · 1P , and the Siegel–Walfisz criterion holds for both f and g This kind of ‘if and only if’ result in the theory of multiplicative functions bears some similarity to (and inspiration from) (1.4) and Theorem 1.2 of [6], and much of the discussion there

More explicit results

The algebra of the Bombieri–Vinogradov hypothesis

Factorizing smooth numbers and using the large sieve

Further thoughts