The lcm(1,2,...,n) as a product of sine values sampled over the points in Farey sequences

TL;DR

This paper presents a concise sine-based proof of Greg Martin's recent formula for the least common multiple of the integers from 1 to n, connecting number theory with trigonometric functions.

Contribution

It offers a new, simplified proof of an existing number theory result using sine functions, enhancing understanding of the formula's structure.

Findings

Established a sine-based proof for the LCM formula

Connected Farey sequences with trigonometric identities

Simplified the understanding of the LCM formula's derivation

Abstract

Recently Greg Martin derived an interesting formula for the least common multiple of {1,2,...,n}. Here, we give an exposition of a concise proof in terms of the sine function.