Nontrivial upper bounds for the least common multiple of an arithmetic progression

Abstract

In this paper, we establish some nontrivial and effective upper bounds for the least common multiple of consecutive terms of a finite arithmetic progression. Precisely, we prove that for any two coprime positive integers [Formula: see text] and [Formula: see text], with [Formula: see text], we have [Formula: see text] where [Formula: see text]. If in addition [Formula: see text] is a prime number and [Formula: see text], then we prove that for any [Formula: see text], we have [Formula: see text], where [Formula: see text]. Finally, we apply those inequalities to estimate the arithmetic function [Formula: see text] defined by [Formula: see text] ([Formula: see text]), as well as some values of the generalized Chebyshev function [Formula: see text].