Some Remarks on Small Values of $τ(n)$

Abstract

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