ABC implies that Ramanujan's tau function misses almost all primes
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2026
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| _version_ | 1866911622355746816 |
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| author | 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 |
| author_facet | 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 |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_29970 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | ABC implies that Ramanujan's tau function misses almost all primes 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 Number Theory 11F11 (Primary), 11F30 (Secondary) 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. |
| title | ABC implies that Ramanujan's tau function misses almost all primes |
| topic | Number Theory 11F11 (Primary), 11F30 (Secondary) |
| url | https://arxiv.org/abs/2603.29970 |