Some Remarks on Small Values of $\tau(n)$

TL;DR

This paper investigates which integers can be values of the Ramanujan tau function, proving that most small integers are not attained and characterizing those that are, while also connecting to conjectures about the function's growth.

Contribution

It synthesizes recent methods to classify small integer values of the tau function and links the results to broader conjectures in the field.

Findings

Most integers with 0<|α|<100 are not values of τ(n)

If τ(n) equals certain small integers, n is square-free with specific prime factors

A strong Atkin-Serre conjecture implies |τ(n)| > 100 for n > 2

Abstract

A natural variant of Lehmer's conjecture that the Ramanujan $\tau$-function never vanishes asks whether, for any given integer $\alpha$, there exist any $n \in \mathbb{Z}^+$ such that $\tau(n) = \alpha$. A series of recent papers excludes many integers as possible values of the $\tau$-function using the theory of primitive divisors of Lucas numbers, computations of integer points on curves, and congruences for $\tau(n)$. We synthesize these results and methods to prove that if $0 < |\alpha| < 100$ and $\alpha \notin T := \{2^k, -24,-48, -70,-90, 92, -96\}$, then $\tau(n) \neq \alpha$ for all $n > 1$. Moreover, if $\alpha \in T$ and $\tau(n) = \alpha$, then $n$ is square-free with prescribed prime factorization. Finally, we show that a strong form of the Atkin-Serre conjecture implies that $|\tau(n)| > 100$ for all $n > 2$.