ABC implies that Ramanujan's tau function misses almost all primes
Year: 2026
Type: preprint
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 \[ S(X):=\#\{\ell\le X:\ \ell\ \text{prime and } |τ(n)|=\ell \text{ for some } n\ge 1\} = O(X^{13/22}), \] 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.
Source: arXiv (Cornell University)
Authors: Angdinata, David Kurniadi, Chen, Evan, Cummins, Chris, Eltschig, Ben, Grubisic, Dejan, Haller, Leopold, Hong, Letong, Kurghinyan, Andranik, Lau, Kenny, Leather, Hugh, Lee, Seewoo, Mahns, Simon, Markosyan, Aram H., Muddana, Rithikesh, Ono, Ken, Patel, Manooshree, Pendharkar, Gaurang, Rathi, Vedant, Schneidman, Alex, Seeker, Volker, Sengupta, Shubho, Sinha, Ishan, Xin, Jimmy, Zhang, Jujian
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7146945806
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