Elementary Proof of Lehmer's Conjecture on Non-vanishing of Tau Function

TL;DR

This paper provides a proof of Lehmer's conjecture, establishing that Ramanujan's tau function never vanishes for positive integers, using additive group structures, the pigeonhole principle, and unique factorization.

Contribution

It offers the first elementary proof of Lehmer's conjecture leveraging basic number theory concepts, avoiding complex analytic methods.

Findings

Tau(n) is non-zero for all n ≥ 1

Elementary methods suffice for proving Lehmer's conjecture

The additive group structure plays a key role in the proof

Abstract

In this paper we prove Lehmer's conjecture on Ramanujan's tau function, namely tau(n) not equal to zero for n >= 1 by investigating the additive group structure attached to tau(n) with the aid of the pigeonhole principle and unique factorization theorem.