Sums with the Mobius function twisted by characters with powerful moduli

TL;DR

This paper improves bounds on sums involving the Mobius function twisted by characters with powerful moduli, using advanced bounds on related L-functions, thereby extending previous work and enhancing understanding of these sums.

Contribution

It introduces new bounds on Mobius function sums twisted by characters with powerful moduli, leveraging improved bounds on associated L-functions and zero-free regions.

Findings

Enhanced bounds on short character sums for nonprincipal characters with powerful moduli

Improved estimates on sums involving the Mobius function twisted by such characters

New results on the size and zero-free regions of related L-functions

Abstract

In their 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 of the Mobius function twisted by characters of the same type, complementing and improving some earlier work of B. Green (2012). To achieve this, we obtain a series of results about the size and zero-free region of $L$-functions with the same class of moduli.