Lehmer's Conjecture on the Non-vanishing of Ramanujan's Tau Function

TL;DR

This paper investigates Lehmer's conjecture that Ramanujan's tau function never vanishes for positive integers, using the additive group structure and unique factorization properties to approach the problem.

Contribution

It introduces a novel approach by examining the additive group structure related to tau(n) and applying unique factorization to analyze the conjecture.

Findings

Preliminary analysis suggests non-vanishing of tau(n) for certain classes of n.

The approach provides new insights into the algebraic structure related to tau(n).

Further work is needed to prove the conjecture completely.

Abstract

In this paper we attempt to prove Lehmer's conjecture on Ramanujan's tau function, namely tau(n) is never zero, for each n larger than zero by investigating the additive group structure attached to tau(n) with the aid of unique factorization theorem.