Asymptotic Improvements Of Lower Bounds For The Least Common Multiples Of Arithmetic Progressions

For relatively prime positive integers u(0) and r, we consider the least common multiple L-n := lcm(u(0), u(1),...,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 -> infinity.