Further improvements of lower bounds for the least common multiples of arithmetic progressions

For relatively prime positive integers $u_0$ and $r$, we consider the arithmetic progression $\{u_k:=u_0+kr\}_{k=0}^n$.

Define $L_n:=\operatorname{lcm}\{u_0, u_1, ..., u_n\}$ and let $a\ge 2$ be any integer. In this paper, we show that for integers $\alpha , r\geq a$ and $n\geq 2\alpha r$, we have

$$L_n\geq u_0r^{\alpha +a-2}(r+1)^n.$$

In particular, letting $a=2$ yields an improvement to the best previous lower bound on $L_n$ (obtained by Hong and Yang) for all but three choices of $\alpha , r\geq 2$.