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

Sid Ali Bousla

Published 2020-04-15Version 1

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 $\theta(x;k,\ell)$.