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 $\le x^{\frac{20}{39}-\delta }$.

54 pages

References in corpus (2)

• Bombieri--Vinogradov and Barban--Davenport--Halberstam type theorems for smooth numbers
• Gowers norms of multiplicative functions in progressions on average

Cited by in corpus (3)

• When does the Bombieri-Vinogradov Theorem hold for a given multiplicative function?
• The Bombieri-Vinogradov theorem for nilsequences
• Zeros of partial sums of $L$-functions