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

David Kurniadi Angdinata, Evan Chen, Chris Cummins, Ben Eltschig, Dejan Grubisic, Leopold Haller, Letong Hong, Andranik Kurghinyan, Kenny Lau, Hugh Leather, Seewoo Lee, Simon Mahns, Aram H. Markosyan, Rithikesh Muddana, Ken Ono, Manooshree Patel, Gaurang Pendharkar, Vedant Rathi, Alex Schneidman, Volker Seeker, Shubho Sengupta, Ishan Sinha, Jimmy Xin, Jujian Zhang

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 \[ S(X):=\#\{\ell\le X:\ \ell\ \text{prime and } |τ(n)|=\ell \text{ for some } n\ge 1\} = O(X^{9/10}\log X), \] 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.

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

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 . Assuming the Conjecture, we prove the stronger upper bound which implies that Ramanujan's tau-function misses a density 1 subset of the primes. We give a heuristic suggesting that should nevertheless be infinite, with predicted order of magnitude The main engine in this note was formalized and produced automatically in Lean/Mathlib by AxiomProver from a natural-language statement of the problem.

Paper Structure

This paper contains 7 sections, 8 theorems, 73 equations.

Table of Contents

  1. Introduction and statement of results
  2. The main engine
  3. The prime-power reduction
  4. Completion via hyperelliptic twists: the small exponents $X_2$ and $X_4$
  5. The proof of Theorem \ref{['thm:main']}
  6. The expected order of $S(X)$
  7. AxiomProver's autonomous Lean verification

Key Result

Theorem 1

Assuming the $abc$ Conjecture, as $X\rightarrow +\infty$ we have

Theorems & Definitions (19)

  • Conjecture : $abc$ Conjecture
  • Theorem 1
  • Remark 2
  • Remark 3
  • Lemma 4
  • Lemma 5
  • Proposition 6: Balakrishnan--Craig--Ono--Tsai BCOT2023
  • Remark 7
  • Proposition 8
  • proof
  • ...and 9 more