Infinitely many prime values of Ramanujan’s tau-function (folklore conjecture)

Determine whether there exist infinitely many primes ℓ such that |τ(n)| = ℓ for some integer n ≥ 1; equivalently, prove that the image of Ramanujan’s tau-function contains infinitely many primes up to sign.

Background

A longstanding folklore conjecture posits the infinitude of primes occurring as absolute values of τ(n). This paper studies the counting function S(X) for such prime values and, assuming the abc Conjecture, proves a strong upper bound showing that τ misses a density 1 subset of primes.

Despite this sparsity, the authors provide a heuristic (using Sato–Tate) suggesting that S(X) should still be infinite, aligning with the folklore conjecture. Prior work of Xiong placed upper-density bounds unconditionally; the present work strengthens these bounds conditionally.