On a Generalisation of a Lehmer Problem

TL;DR

This paper generalizes the Lehmer problem concerning the distribution of modular inverses in arithmetic progressions, employing bounds of multiplicative character sums to improve previous results.

Contribution

It introduces a broader framework for the Lehmer problem and enhances existing bounds by using multiplicative character sum estimates instead of Kloosterman sums.

Findings

Improved bounds on the distribution of modular inverses

Extended the Lehmer problem to a more general setting

Enhanced results using multiplicative character sum techniques

Abstract

We consider a generalisation of the classical Lehmer problem about the distribution of modular inverses in arithmetic progression, introduced by E. Alkan, F. Stan and A. Zaharescu. Using bounds of sums of multiplicative characters instead of traditionally applied to this kind of problem Kloosterman sums, we improve their results in several directions.