Asymptotic behavior of the least common multiple of consecutive arithmetic progression terms

TL;DR

This paper investigates the asymptotic growth of the logarithm of the least common multiple of terms in a specific arithmetic progression, establishing a linear growth rate with a constant depending on progression parameters.

Contribution

It provides a new asymptotic formula for the logarithm of the LCM of consecutive arithmetic progression terms, extending understanding of their growth behavior.

Findings

Log of LCM grows linearly with n

The growth rate constant A depends on l, m, and a

Asymptotic formula includes an o(n) error term

Abstract

Let $l$ and $m$ be two integers with $l>m\ge 0$, and let $a$ and $b$ be integers with $a\ge 1$ and $a+b\ge 1$. In this paper, we prove that $\log {\rm lcm}_{mn<i\le ln}\{ai+b\} =An+o(n)$, where $A$ is a constant depending on $l, m$ and $a$.