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 $a$ and $b$, with $b\geq 2$, we have \[\mathrm{lcm}\left(a,a+b,\dots,a+nb\right) \leq \left(c_1\cdot b\log b\right)^{n+\left\lfloor \frac{a}{b}\right\rfloor}~~~~(\forall n\geq b+1),\] where $c_1=41.30142$. If in addition $b$ is a prime number and $a<b$, then we prove that for any $n\geq b+1$, we have $\mathrm{lcm}\left(a,a+b,\dots,a+nb\right) \leq \left(c_2\cdot b^{\frac{b}{b-1}}\right)^n$, where $c_2=12.30641$. Finally, we apply those inequalities to estimate the arithmetic function $M$ defined by $M(n):=\frac{1}{\varphi(n)}\sum_{\substack{1\leq\ell\leq n \\ \ell \wedge n=1}}\frac{1}{\ell}$ ($\forall n \geq 1$), as well as some values of the generalized Chebyshev function $\theta(x;k,\ell)$.