Lower bounds for lcm of integers in arithmetic progression

Abstract

The bounds of the least common multiple for the finite sequences in arithmetic progressions are interesting questions. The sum of von Mangoldt function of the first n positive integers is exactly the logarithm of their least common multiple. Such questions are thus closely related to Prime Number Theorem. However, we may also be interested on the effective uniform bound, and this is already answered by Hanson and Nair thirty years ago. In this talk, we shall consider positive integers in arithmetic progressions and discuss some advancement on such estimate.