Some remarks on small values of $\tau (n)$

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 $0< \left| \alpha \right| < 100$ and $\alpha \notin T := \{2^k, -24,-48, -70,-90, 92, -96\}$, then $\tau (n) \ne \alpha$ 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 $\left| \tau (n) \right| > 100$ for all $n > 2$.