Bombieri-Vinogradov for multiplicative functions, and beyond the x1/2-barrier

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 <= x(20/39-delta). (C) 2019 Elsevier Inc. All rights reserved.