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

Daniel M. Kane; Scott Duke Kominers

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Article | Canadian Mathematical Bulletin

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

by Daniel M. Kane and Scott Duke Kominers

Abstract

For relatively prime positive integers and r, we consider the least common multiple L_n:=\mathop{\textrm{lcm}}(u_0,u_1,\dots, u_n) of the finite arithmetic progression \{u_k:=u_0+kr\}_{k=0}^n. We derive new lower bounds on L_n that 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.

Keywords: Mathematical Methods;

Format: Print

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Citation:

Kane, Daniel M., and Scott Duke Kominers. "Asymptotic Improvements of Lower Bounds for the Least Common Multiples of Arithmetic Progressions." Canadian Mathematical Bulletin 57, no. 3 (September 2014): 551–561.

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

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