When does the Bombieri-Vinogradov Theorem hold for a given multiplicative function?

TL;DR

This paper investigates the conditions under which the Bombieri-Vinogradov Theorem applies to specific multiplicative functions, linking it to the Siegel-Walfisz criterion and prime restrictions.

Contribution

It establishes a precise equivalence between the theorem's validity for multiplicative functions and certain criteria involving prime restrictions and the Siegel-Walfisz condition.

Findings

The theorem holds for both functions if and only if the Siegel-Walfisz criterion holds for each.

The theorem's applicability is characterized by its validity on prime-restricted functions.

The results connect the Bombieri-Vinogradov Theorem to classical number theory criteria.

Abstract

Let $f$ and $g$ be $1$-bounded multiplicative functions for which $f*g=1_{.=1}$. 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.