Variations of Lehmer's Conjecture for Ramanujan's tau-function

TL;DR

This paper proves that Ramanujan's tau-function does not take certain small integer values for n > 1, using advanced number theory techniques, thus extending our understanding of its non-vanishing properties.

Contribution

The authors establish new non-vanishing results for Ramanujan's tau-function for specific small integers, employing methods involving Lucas sequences, hyperelliptic curves, and Thue equations.

Findings

Tau(n) does not equal ±1, ±3, ±5, ±7, or ±691 for n > 1.

The proof simplifies in cases where Ramanujan's congruences apply.

The work connects tau-function properties with Galois representations and Diophantine equations.

Abstract

We consider natural variants of Lehmer's unresolved conjecture that Ramanujan's tau-function never vanishes. Namely, for $n>1$ we prove that $$\tau(n)\not \in \{\pm 1, \pm 3, \pm 5, \pm 7, \pm 691\}.$$ This result is an example of general theorems for newforms with trivial mod 2 residual Galois representation, which will appear in forthcoming work of the authors with Wei-Lun Tsai. Ramanujan's well-known congruences for $\tau(n)$ allow for the simplified proof in these special cases. We make use of the theory of Lucas sequences, the Chabauty-Coleman method for hyperelliptic curves, and facts about certain Thue equations.