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 $0<|\alpha |<100$ 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 $|\tau (n)|>100$ for all $n>2$.

To appear in Archiv der Mathematik