Asymptotic Improvements of Lower Bounds for the Least Common Multiples of Arithmetic Progressions

TL;DR

This paper develops improved asymptotic lower bounds for the least common multiple of finite arithmetic progressions, especially when the initial term or the number of terms is large, with sharp bounds when the common difference is prime.

Contribution

It introduces new lower bounds on the LCM of arithmetic progressions that are asymptotically sharper than previous bounds, particularly for large parameters.

Findings

New lower bounds improve previous results for large $u_0$ or $n$.

Bounds are nearly sharp as $n$ approaches infinity when $r$ is prime.

Best bounds are sharp up to a factor of $n+1$ for specific $u_0$ values.

Abstract

For relatively prime positive integers $u_0$ and $r$, we consider the least common multiple $L_n:=\mathrm{lcm}(u_0,u_1,\ldots, u_n)$ of the finite arithmetic progression $\{u_k:=u_0+kr\}_{k=0}^n$. We derive new lower bounds on $L_n$ which improve upon those obtained previously when either $u_0$ or $n$ is large. When $r$ is prime, our best bound is sharp up to a factor of $n+1$ for $u_0$ properly chosen, and is also nearly sharp as $n\to\infty$.