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

Assuming the $abc$ Conjecture, this paper proves that Ramanujan's tau-function takes prime values for only a set of primes with density zero, specifically establishing an upper bound of $O(X^{9/10}\log X)$ for the count of such primes up to $X$ , while also providing a heuristic prediction for their infinite yet sparse occurrence.

Original authors: 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

Published 2026-04-01

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: 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

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a magical machine called Ramanujan's Tau-Function. You feed it a number (like 1, 2, 3...), and it spits out a result. For a long time, mathematicians have been fascinated by this machine, asking two big questions:

Does it ever break? (Can it output zero?)
Does it ever spit out a "Prime Number"? (Prime numbers are the building blocks of math, like 2, 3, 5, 7, 11...)

For decades, people believed the machine might spit out infinitely many prime numbers. But this new paper, written by a massive team of humans and an AI, says: "Actually, under a very famous mathematical rule, this machine almost never spits out primes. It misses almost all of them."

Here is the story of how they figured it out, using simple analogies.

1. The Mystery of the Missing Primes

Think of the prime numbers as a giant, endless ocean of islands. The Tau-function is a boat that sails across this ocean.

The Old Belief: Many sailors thought the boat would eventually visit every island (or at least an infinite number of them).
The New Discovery: This paper proves that if you accept a specific rule of the universe (the abc Conjecture), the boat is actually a terrible navigator. It sails past 99.9% of the islands. It only stops at a tiny, tiny speck of the ocean.

2. The "abc Conjecture": The Universe's Speed Limit

To make this proof work, the authors had to assume the abc Conjecture is true.

The Analogy: Imagine the abc Conjecture is a cosmic speed limit or a law of physics. It says that if you add two numbers together to get a third, the "ingredients" (the prime factors) of those numbers can't be too small compared to the final result.
Why it matters: The authors used this "speed limit" to show that the mathematical equations governing the Tau-function are so restrictive that they simply cannot land on most prime numbers. It's like trying to park a massive truck in a tiny parking spot; the laws of physics (the abc Conjecture) say it's just not going to fit.

3. The "Hyperbolic" Hunt

The paper dives deep into the math, looking at specific shapes called hyperelliptic curves.

The Analogy: Imagine the Tau-function is a hiker trying to find a specific type of wildflower (a prime number) in a vast, twisted mountain range. The hiker can only walk on specific paths (mathematical equations).
The authors proved that under the "speed limit" rule, the paths that lead to prime flowers are incredibly narrow and rare. Most of the time, the hiker walks right past the flowers without seeing them. They calculated that the number of primes found grows so slowly that, compared to the total number of primes, it's basically zero.

4. The Twist: It Might Still Happen (Just Rarely)

Even though the paper says the machine "misses almost all primes," the authors don't think it never finds one.

The Heuristic (The Guess): They did a "back-of-the-napkin" calculation suggesting that the machine does find primes, but they are as rare as finding a specific grain of sand on all the beaches on Earth.
The Prediction: They predict the number of primes found follows a very specific, slow pattern (like $X^{1/11}$ ). It's not zero, but it's so sparse that for all practical purposes, the machine is "missing" them.

5. The AI Co-Author: The "Robot Mathematician"

This is perhaps the most exciting part for the future of science.

The Human-AI Team: The paper lists 24 authors. Some are humans, but one is an AI called AxiomProver.
The Job: The humans wrote the problem in plain English. They told the AI: "Here is the problem. Here are the rules. Please prove it."
The Result: The AI didn't just check the math; it wrote the proof in a computer language called Lean. It translated the human idea into a rigorous, computer-verifiable code.
The Analogy: Imagine a human architect draws a blueprint for a bridge. Then, a robot takes that blueprint, calculates every single stress point, and builds the bridge in a simulation to prove it won't collapse. The human then writes a story about the bridge for other people to read.

The Big Takeaway

This paper is a landmark because it combines deep human intuition with AI's ability to handle complex logic.

The Result: Ramanujan's Tau-function is a "prime-hater." It skips almost every prime number in existence.
The Method: They used a famous rule (abc Conjecture) to prove that the math simply doesn't allow for many prime outputs.
The Future: An AI was able to take a natural language description of a hard math problem and generate a formal proof. This suggests that in the future, AI might become a standard partner for mathematicians, handling the heavy lifting of verification while humans focus on the big ideas.

In short: The machine exists, but it's very picky about which primes it likes, and an AI helped us prove exactly how picky it is.

1. Problem Statement

The paper addresses the distribution of prime values in the image of Ramanujan's tau-function, denoted as $\tau(n)$ . This function arises as the Fourier coefficients of the normalized weight 12 cusp form $\Delta(z)$ on $SL_2(\mathbb{Z})$ :
$\Delta(z) = q \prod_{n=1}^\infty (1-q^n)^{24} = \sum_{n=1}^\infty \tau(n)q^n$
where $q = e^{2\pi i z}$ .

While Lehmer's conjecture posits that $\tau(n) \neq 0$ for all $n \geq 1$ , a related "folklore" conjecture suggests that the set of values $\{|\tau(n)| : n \geq 1\}$ contains infinitely many primes. Previous work by Xiong established that the set of such primes has an upper density of at most $2/11$ within the set of all primes.

The Core Question: Does the tau-function take prime values infinitely often, and if so, how dense is this set of primes? Specifically, the authors investigate the counting function:
$S(X) := \#\{\ell \leq X : \ell \text{ is prime and } |\tau(n)| = \ell \text{ for some } n \geq 1\}$

2. Methodology

The authors employ a combination of analytic number theory, modular form theory, and conditional assumptions to derive upper bounds for $S(X)$ .

A. Theoretical Framework

Hecke Relations and Parity: The authors utilize the multiplicative properties of $\tau(n)$ and Deligne's bound ( $|\tau(p)| \leq 2p^{11/2}$ ). Crucially, they rely on Proposition 8, which states that $\tau(n)$ is odd if and only if $n$ is an odd square. Consequently, if $|\tau(n)|$ is an odd prime, $n$ must be of the form $p^{2m}$ .
Reduction to Hyperelliptic Curves:
- For small exponents ( $m=1, 2$ ), the condition $|\tau(p^{2m})| = \ell$ is transformed into finding integer points on specific hyperelliptic curves.
- For $m=1$ ( $n=p^2$ ): $\tau(p^2) = \tau(p)^2 - p^{11} = \pm \ell$ leads to the curve $y^2 = x^{11} \pm \ell$ .
- For $m=2$ ( $n=p^4$ ): The relation leads to the curve $u^2 = 5x^{22} \pm 4\ell$ .
The abc Conjecture: The central engine of the proof is the abc Conjecture (Masser-Oesterlé). The authors use this to bound the number of integer points on the aforementioned curves where the "radical" of the equation is small relative to the magnitude of the terms.
Decomposition of $S(X)$ : The counting function $S(X)$ $S (X)$ is split into contributions from different prime-power exponents $k$ $k$ in $n=p^k$ $n = p^{k}$ :
- $k \geq 3$ : Bounded using Xiong's existing results.
- $k = 1, 2$ : Bounded using the new analytic estimates derived from the abc Conjecture.

B. Analytic Estimates

The authors define error terms $E_2(X)$ and $E_4(X)$ representing the number of integer points on the curves $y^2 = x^{11} \pm \ell$ and $u^2 = 5x^{22} \pm 4\ell$ respectively, within specific ranges. They prove (assuming the abc Conjecture) that:

$E_2(X) \ll X^{4/9+\eta}$
$E_4(X) \ll X^{1/5+\eta}$
These bounds are significantly tighter than the trivial bounds, allowing the dominant term in the final estimate to be controlled.

3. Key Contributions and Results

Main Theorem (Theorem 1)

Assuming the abc Conjecture, the authors prove the following asymptotic upper bound for the number of prime values:
$S(X) = O\left( \frac{X^{9/10}}{\log X} \right)$
Implication: Since the Prime Number Theorem states that the total number of primes $\pi(X) \sim X/\log X$ , the ratio $S(X)/\pi(X) \to 0$ as $X \to \infty$ .
Conclusion: Ramanujan's tau-function misses a density 1 subset of the primes. In other words, almost all primes are not in the image of $\tau(n)$ .

Heuristic Prediction (Section 4)

Despite the upper bound proving the set is "thin" (density zero), the authors provide a heuristic argument suggesting $S(X)$ is still infinite.

Using the Sato-Tate Conjecture (proven by Clozel, Harris, Shepherd-Barron, and Taylor), they argue that the angles $\theta_p$ associated with $\tau(p)$ are equidistributed.
This leads to a predicted order of magnitude:
$S(X) \asymp \frac{X^{1/11}}{(\log X)^2}$
This suggests that while prime values exist, they are extremely sparse compared to the total number of primes.

Generalization (Remark 3)

The result is not limited to $\tau(n)$ . The authors note that the same logic applies to the coefficients of any normalized Hecke eigenform on $SL_2(\mathbb{Z})$ with integer coefficients.

4. Significance and AI Integration

Mathematical Significance

Refinement of Density Bounds: This work significantly improves upon Xiong's previous bound of density $2/11$ , pushing the upper bound to density 0 (conditional on abc).
Connection to Diophantine Geometry: The proof elegantly bridges the gap between the arithmetic of modular forms and the Diophantine properties of hyperelliptic curves via the abc Conjecture.
Clarification of "Folklore": It resolves the tension between the existence of prime values (Lehmer found one) and their scarcity, showing they are vanishingly rare.

AI and Formal Verification (Section 5)

A unique feature of this paper is the role of AxiomProver, an AI system developed by the authors.

Autonomous Formalization: The authors provided natural language statements of the problem and relevant lemmas to AxiomProver.
Output: The AI autonomously generated a complete formal proof in Lean 4 (using the Mathlib library) assuming the abc Conjecture and specific results from Xiong's paper.
Verification: The human authors then translated the Lean code back into a human-readable narrative for this paper.
Significance: This demonstrates a new paradigm in mathematical research where AI can handle the rigorous, mechanical steps of formal verification and proof construction, allowing humans to focus on high-level strategy and exposition.

Summary

The paper proves that, conditional on the abc Conjecture, Ramanujan's tau-function takes prime values so rarely that the set of such primes has natural density zero. While the function likely produces infinitely many primes, they are exponentially sparse. The work is notable both for its deep number-theoretic results and for its pioneering use of AI to generate and verify the formal proof.