2020/04/15 by Sid Ali Bousla, Bousla, Sid Ali
Computer Science · Mathematics · #11A05 #11B25 (Primary) #Analytic Number Theory Research #FOS: Mathematics #Graph Labeling and Dimension Problems #Limits and Structures in Graph Theory #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2004.07335
openalex publication_date 2020/04/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we establish some nontrivial and effective upper bounds for\nthe least common multiple of consecutive terms of a finite arithmetic\nprogression. Precisely, we prove that for any two coprime positive integers a\nand b, with b\≥ 2, we have \
mathrmlcm
left(a,a+b,
dots,a+nb
right)\n
leq
left(c1
cdot b
log b
right)^n+
left
lfloor\n
fracab
right
rfloor~~~~(
forall n
geq b+1), where c1=41.30142. If in\naddition b is a prime number and a<b, then we prove that for any n\≥\nb+1, we have \lcm\(a,a+b,\…,a+nb\) \≤ \(c2\⋅\nb\(b)/(b-1)\)n, where c2=12.30641. Finally, we apply those\ninequalities to estimate the arithmetic function M defined by\nM(n):=\(1)/(\φ(n))\∑_ substack1\≤\ℓ\≤ n \ℓ wedge\nn=1\(1)/(\ℓ) (\∀ n \≥ 1), as well as some values of the\ngeneralized Chebyshev function \θ(x;k,\ℓ).\n