Sums with the Möbius function twisted by characters with powerful moduli

Abstract

In a recent work, the authors (2016) have combined classical ideas of A. G. Postnikov (1956) and N. M. Korobov (1974) to derive improved bounds on short character sums for certain nonprincipal characters with powerful moduli. In the present paper, these results are used to bound sums with the Möbius function twisted by characters of the same type, which complements and improves some earlier work of B. Green (2012). To achieve this, we obtain a series of results about the size and zero-free region of L L -functions with the same class of moduli.