On some values which do not belong to the image of Ramanujan's tau-function

TL;DR

This paper investigates the values of Ramanujan's tau function, proving that it does not take certain multiples of primes less than 1000, extending previous results and identifying 14 exceptional cases.

Contribution

It extends known non-vanishing results of the tau function to larger primes and more complex multiples, covering primes less than 1000 and identifying specific exceptions.

Findings

Tau(n) does not equal ±ℓ, ±2ℓ, ±4ℓ, ±8ℓ for any n ≥ 1 with 14 exceptions

Proves non-membership of tau values in certain arithmetic progressions for primes < 1000

Builds on previous work with primes < 100 and extends to larger primes and more multiples.

Abstract

Lehmer conjectured that Ramanujan's tau function never vanishes. As a variation of this conjecture, it is proved that \begin{equation*} \tau(n)\neq \pm \ell, \pm 2\ell, \pm 2\ell^2, \end{equation*} where $\ell<100$ is an odd prime, by Balakrishnan, Ono, Craig, Tsai and many people. We have proved that \begin{equation*} \tau(n)\neq \pm \ell, \pm 2\ell, \pm 4\ell, \pm 8\ell \end{equation*} for any $n\geq 1$ except 14 cases, where $\ell<1000$ is an odd prime.