ABC implies that Ramanujan's tau function misses almost all primes

TL;DR

Under the assumption of the abc conjecture, the paper proves that Ramanujan's tau function misses almost all primes, providing an upper bound on the primes it attains and suggesting it may still take infinitely many prime values.

Contribution

The paper establishes a new conditional upper bound on the prime values of Ramanujan's tau function assuming the abc conjecture, showing it misses a density 1 subset of primes.

Findings

Proves S(X) = O(X^{13/22}) assuming abc conjecture.

Shows Ramanujan's tau function misses almost all primes.

Heuristic predicts S(X) should be infinite with order ~ C X^{1/11}/(log X)^2.

Abstract

Lehmer conjectured that Ramanujan's tau-function never vanishes. In a related direction, a folklore conjecture asserts that infinitely many primes arise as absolute values of Ramanujan's tau-function. Recently, Xiong showed that these prime values form a subset of the primes with density at most $2/11$. Assuming the $abc$ Conjecture, we prove the stronger upper bound \[ S(X):=\#\{\ell\le X:\ \ell\ \text{prime and } |\tau(n)|=\ell \text{ for some } n\ge 1\} = O(X^{13/22}), \] which implies that Ramanujan's tau-function misses a density 1 subset of the primes. We give a heuristic suggesting that $S(X)$ should nevertheless be infinite, with predicted order of magnitude \[ S(X)\asymp \frac{C X^{\frac{1}{11}}}{(\log X)^2}. \] The main engine in this note was formalized and produced automatically in Lean/Mathlib by AxiomProver from a natural-language statement of the problem.