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 $2/11$. 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.

• The paper refines density bounds for prime values of Ramanujan's tau function under the abc conjecture, proving its image comprises a zero density subset of all primes.
• It utilizes prime power criteria, Hecke relations, Deligne's bounds, and hyperelliptic curve analysis to derive a conditional upper bound of O(X^(9/10) log X).
• The results imply that despite the tau function taking infinitely many values, almost all primes are missed, deepening our understanding of modular form arithmetic.

Density of Prime Values in Ramanujan’s Tau Function Under ABC

Introduction

The paper "ABC implies that Ramanujan's tau function misses almost all primes" (2603.29970) investigates the distribution of prime values taken by Ramanujan’s tau function $\tau(n)$, specifically addressing the density of primes $\ell$ such that $|\tau(n)| = \ell$ for some $n$. The central result is a substantial improvement of previous upper bounds on the density of such primes, contingent upon the $abc$ Conjecture. The authors provide a rigorous conditional analysis that, assuming $abc$, the set of prime values occupied by $\tau(n)$ comprises a density zero subset of all primes.

Background and Context

Ramanujan's tau function emerges as the Fourier coefficients of the discriminant modular form $\Delta(z)$, a normalized weight 12 cusp form on $\mathrm{SL}_2(\mathbb{Z})$. Lehmer conjectured $\tau(n)\neq0$ for all $\ell$0, a question that has driven much subsequent research. A related folklore conjecture posits that $\ell$1 takes infinitely many prime values (up to sign). Previous work, including Xiong’s recent results, established that the density of such primes is at most $\ell$2, based on residue class restrictions modulo $\ell$3.

The $\ell$4 Conjecture, as formulated by Masser and Oesterlé, asserts that for coprime $\ell$5 with $\ell$6, the maximum of $\ell$7 is tightly controlled by the radical of $\ell$8. This conjecture has deep implications for Diophantine equations and thus informs many results about the distribution of arithmetic function values.

Main Results

The key numerical claim in this paper is the following conditional upper bound:

$\ell$9

This refinement, assuming the $|\tau(n)| = \ell$0 Conjecture, supersedes Xiong’s previous density bound and implies that the tau function misses a density 1 subset of the primes:

$|\tau(n)| = \ell$1

where $|\tau(n)| = \ell$2 is the prime counting function. Thus, the collection of primes in the image of $|\tau(n)| = \ell$3 is, asymptotically, negligible.

The authors further provide a heuristic, grounded in Sato-Tate theory and Chebyshev polynomial behavior, suggesting that while the collection is sparse, it remains infinite. Predicting the asymptotic:

$|\tau(n)| = \ell$4

where $|\tau(n)| = \ell$5 is an explicit constant, and the dominant contribution arises from $|\tau(n)| = \ell$6 in the expansion of $|\tau(n)| = \ell$7 for odd prime $|\tau(n)| = \ell$8 and integer $|\tau(n)| = \ell$9.

Technical Approach

The proof leverages several deep arithmetic results:

• Prime Power Criteria: The main reduction relies on established results regarding the structure of prime values attained by $n$0, specifically those arising in prime-power exponents, coupled with parity constraints.
• Hecke Relations & Deligne's Bound: The analytic and algebraic structure of the coefficients is exploited through Hecke multiplicativity and Deligne's bounds on Fourier coefficients, yielding explicit forms for $n$1.
• Hyperelliptic Curve Analysis: For low exponents ($n$2), the authors reduce the problem to integer point counts on specific hyperelliptic curves. The complexity of these counts is managed via interval estimates and bounds on the range of possible $n$3 and $n$4 or $n$5 variables.
• Application of $n$6 Conjecture: The $n$7 Conjecture is invoked to bound solutions to $n$8 and $n$9 for small $abc$0. The main technical engine consists of bounding the number of integer points near the relevant curves, showing that they contribute negligibly as $abc$1.

The proof is formalized in Lean, via AxiomProver, attesting to both the correctness and the adaptability of the protocol for automated mathematics using current theorem proving infrastructures.

Implications for Number Theory

This result decisively establishes, conditionally, that the image of the tau function on primes is a thin set. This outcome has consequences for the theory of modular forms and their arithmetic properties, affording an explicit illustration where rich algebraic structure does not translate into dense images among primes. The methods further generalize to normalized Hecke eigenforms on $abc$2 with integer coefficients, underscoring a universality in the sparsity phenomenon.

Additionally, this paper demonstrates the feasibility and utility of formal methods in verifying complex analytic number theory proofs. The successful deployment of AxiomProver and Lean illustrates ongoing progress towards machine-assisted and eventually fully automated mathematical theorem proving, with significant implications for future research cycles, formalization standards, and mathematical publishing.

Speculation and Future Directions

The heuristic prediction alludes to potentially infinite—but exceedingly rare—prime values among tau coefficients. While the $abc$3 Conjecture remains unproven, the demonstrated protocols suggest that verifying such density claims under further conditional hypotheses (e.g., more general modular forms, different correlation structures) is tractable.

The integration of formal proof assistants in nontrivial arithmetic research promises to catalyze further development in both theory and software. As large-scale formalization efforts reach a critical mass, the interplay between analytic estimates, algebraic constraints, and automated symbolic reasoning will become essential in constructing and validating deep conjectural frameworks in arithmetic geometry and computational number theory.

Conclusion

The paper establishes, under the $abc$4 Conjecture, that Ramanujan's tau function omits almost all primes, rigorously quantifying the asymptotic scarcity of prime values in its image. The technical analysis synthesizes classical modular form theory, advanced analytic estimates, and formal proof validation. The implication extends both theoretically—clarifying the arithmetic of modular forms—and practically, exemplifying the integration of automated theorem proving in contemporary mathematical research (2603.29970).