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On the number of exceptional intervals to the prime number theorem in short intervals

This paper establishes an explicit relation between zero density estimates and bounds on the exceptional set for the prime number theorem in short intervals, enabling the derivation of improved upper bounds for almost all $x$ when $\theta > \frac{2}{15}$ through the application of recent zero density estimates and minimal computer assistance.

Original authors: Ayla Gafni, Terence Tao

Published 2026-05-27

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ayla Gafni, Terence Tao

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

The Big Picture: Counting Primes in a Rush

Imagine you are a librarian trying to count books (which represent prime numbers) on a very long shelf. You know a general rule: if you look at a huge section of the shelf, the number of books you find is very predictable. This is the famous Prime Number Theorem.

But what if you only have time to look at a tiny, short section of the shelf? Maybe just a few inches?

The Expectation: Mathematicians believe that even in these tiny sections, the number of books should still follow the predictable pattern. If you expect to find 10 books in a short stretch, you should find roughly 10.
The Problem: Sometimes, nature is messy. In some very specific, tiny sections, the count might be wildly off. You might find 2 books when you expected 10, or 18 when you expected 10. These are the "Exceptional Intervals."

The paper asks: How often do these "messy" intervals happen?

The Analogy: The Foggy Window

Think of the distribution of prime numbers as a clear picture. However, there is a "fog" (mathematical uncertainty) that sometimes obscures the picture in short intervals.

The "Fog" (Zeroes of the Zeta Function): In the background of this math, there is a complex object called the Riemann Zeta function. It has special points called "zeroes." You can think of these zeroes as the source of the fog. The more zeroes there are in certain dangerous zones, the thicker the fog gets, and the more likely it is that our prime number count will be wrong in a short interval.
The Goal: The authors want to measure exactly how much "fog" exists. If they can prove the fog is thin enough, they can guarantee that the "messy" intervals are extremely rare.

The New Discovery: A Better Map

Before this paper, mathematicians had some maps showing where the fog was thick, but they weren't very detailed. They knew that for very short intervals, the fog was too thick to be sure of the count.

Gafni and Tao (the authors) did two main things:

They built a new, more precise translator.
They created a mathematical formula that acts like a translator. It takes the latest, most accurate maps of the "fog" (known as Zero Density Estimates) and instantly converts them into a limit on how many "messy" intervals can exist.
- Analogy: Imagine you have a weather satellite that tells you exactly how much rain is falling in a storm. The authors built a machine that takes that rain data and instantly tells you exactly how many puddles will form on the sidewalk.
They used a computer to crunch the numbers.
Because the "fog" maps are incredibly complex (involving many different curves and ranges), the authors used a computer to run the numbers through their new formula.
- The Result: They found that the "messy" intervals are even rarer than we previously thought. They provided specific numbers showing that for almost all short intervals, the Prime Number Theorem holds true.

What They Found (The "So What?")

The paper doesn't claim to solve the mystery of why primes are distributed this way (that's still a huge mystery). Instead, it solves a specific side problem: How bad can the exceptions get?

The "Exceptional Set": This is the collection of all the "bad" intervals where the count is wrong.
The Finding: The authors proved that this "Exceptional Set" is incredibly small. In fact, if you picked a random short interval, the odds of it being "messy" are so low that it's almost impossible to find one.
The Connection: They showed that if we can prove the "fog" (the zeroes) is thinner than a certain amount, then the "messy" intervals disappear almost entirely.

Why This Matters (In Simple Terms)

Think of it like a quality control check on a factory line.

The Factory: Produces prime numbers.
The Defect: A short interval where the count is wrong.
The Old Method: We knew defects happened, but we didn't know exactly how rare they were.
The New Method: Gafni and Tao used the best available data on the factory's machinery (the Zero Density Estimates) to calculate the exact maximum number of defects possible.

They found that the factory is running much cleaner than we thought. While we can't say the defects never happen, they are so rare that for all practical purposes, the Prime Number Theorem works perfectly in short intervals.

Summary

This paper is a bridge. It connects two complex areas of math:

Zero Density Estimates: How many "foggy" points exist in the background.
Exceptional Intervals: How often the prime number count goes wrong in short bursts.

By building a stronger bridge between them, the authors showed that the "fog" is thin enough to guarantee that the Prime Number Theorem works almost everywhere, even in very short intervals. They didn't find a new type of prime; they just proved that the "bad" ones are vanishingly rare.

Technical Summary: On the Number of Exceptional Intervals to the Prime Number Theorem in Short Intervals

Problem Statement
The paper addresses the validity of the Prime Number Theorem (PNT) in short intervals. Specifically, for a fixed exponent $0 < \theta \le 1$ , the conjecture is that the sum of the von Mangoldt function $\Lambda(n)$ over an interval $[x, x+x^\theta]$ satisfies $\sum_{x<n\le x+x^\theta} \Lambda(n) \sim x^\theta$ as $x \to \infty$ . While this is known to hold for all $x$ when $\theta > 17/30$ (based on recent zero-density estimates) and for almost all $x$ when $\theta > 2/15$ , the behavior fails for certain "exceptional" values of $x$ in the range $2/15 < \theta \le 17/30$ .

The central problem is to quantify the size of this exceptional set $E_\delta(X, \theta)$ , defined as the set of $x \in [X, 2X]$ where the PNT fails by a relative error $\delta$ . The authors aim to determine the exponent $\mu(\theta)$ such that the Lebesgue measure of the exceptional set satisfies $|E_\delta(X, \theta)| \ll X^{\mu(\theta)+o(1)}$ . A value of $\mu(\theta) < 1$ implies the PNT holds for almost all $x$ , while $\mu(\theta) = -\infty$ implies it holds for all $x$ .

Methodology
The authors establish a rigorous connection between zero-density estimates of the Riemann zeta function and the bounds on the exceptional set exponent $\mu(\theta)$ .

Explicit Formula and Moment Estimates: The proof strategy relies on the explicit formula for $\sum \Lambda(n)$ . The authors decompose the error term into contributions from the non-trivial zeros $\rho = \beta + i\gamma$ of $\zeta(s)$ . They analyze the magnitude of these contributions using $L^\infty$ , $L^2$ , and $L^4$ moment estimates.
Zero Density and Additive Energy:
- The standard zero-density theorem provides bounds on $N(\sigma, T)$ , the number of zeros with real part $\ge \sigma$ and imaginary part $\le T$ . The exponent $A(\sigma)$ is defined such that $N(\sigma, T) \ll T^{A(\sigma)(1-\sigma)+\varepsilon}$ .
- Crucially, the authors incorporate bounds on the additive energy of zeros, denoted $N^*(\sigma, T)$ , which counts quadruples of zeros $(\rho_1, \dots, \rho_4)$ such that $|\gamma_1 + \gamma_2 - \gamma_3 - \gamma_4| \le 1$ . The associated exponent is $A^*(\sigma)$ .
Derivation of Bounds: By applying the explicit formula and bounding the resulting sums via the $L^2$ $L^{2}$ and $L^4$ $L^{4}$ norms, the authors derive two competing upper bounds for the contribution of zeros in a specific real-part range $[\sigma, \sigma+\epsilon]$ $[σ, σ + ϵ]$ :
- A bound derived from the second moment involving $A(\sigma)$ : $\mu_{2,\sigma}(\theta) = (1-\theta)(1-\sigma)A(\sigma) + 2\sigma - 1$ .
- A bound derived from the fourth moment involving $A^*(\sigma)$ : $\mu_{4,\sigma}(\theta) = (1-\theta)(1-\sigma)A^*(\sigma) + 4\sigma - 3$ .
Optimization: The final bound for $\mu(\theta)$ is obtained by taking the infimum over $\varepsilon$ and the supremum over $\sigma$ of the minimum of these two quantities, subject to the constraint that the zero density is sufficiently high ( $A(\sigma) \ge \frac{1}{1-\theta} - \varepsilon$ ).

Key Contributions

Theorem 1.2 (General Bound): The paper establishes a general formula relating $\mu(\theta)$ to the zero-density exponent $A(\sigma)$ . This recovers previous results, such as the condition for the PNT to hold for all $x$ ( $\theta > 1 - 1/A_0$ ) and for almost all $x$ ( $\theta > 1 - 2/A_0$ ).
Theorem 1.3 (Refined Bound): The primary theoretical contribution is the inclusion of the additive energy exponent $A^*(\sigma)$ . The authors prove that $\mu(\theta)$ is bounded by the minimum of the second-moment and fourth-moment derived bounds. This refinement allows for tighter constraints on the exceptional set, particularly in ranges where $A^*(\sigma)$ is significantly smaller than $3A(\sigma)$ .
Explicit Relation to Recent Estimates: The authors provide a framework to directly apply the most recent zero-density estimates (specifically those by Guth and Maynard, and Tao–Trudgian–Yang) to bound the exceptional set. This is achieved through a combination of analytic derivation and computer-assisted numerical evaluation.

Results

Unconditional Bounds: Using the current best unconditional bounds for $A(\sigma)$ $A (σ)$ and $A^*(\sigma)$ $A^{*} (σ)$ (summarized in Tables 1 and 2), the authors compute explicit upper bounds for $\mu(\theta)$ $μ (θ)$ .
- For $\theta > 17/30$ , the exceptional set is empty (PNT holds for all $x$ ).
- For $\theta > 2/15$ , the exceptional set has density zero.
- In the intermediate range, the paper provides specific numerical bounds. For example, at $\theta = 17/30$ , they show $\mu(17/30) \le 7/12$ .
Comparison with Previous Work: The authors note that their method recovers the bounds of Bazzanella and Perelli but improves upon them in certain ranges by utilizing the additive energy estimates. They explicitly state that they could not replicate a specific claim in a previous work (Bazzanella [4]) regarding the range $1/6 < \theta \le 1/2$ , identifying a gap in the previous argument related to the Jutila density estimate.
Hypothetical Scenarios: The paper analyzes the bounds under the Density Hypothesis (DH), Lindelöf Hypothesis (LH), and Riemann Hypothesis (RH). Under RH, $\mu(\theta) = -\infty$ for $\theta > 1/2$ . Under DH, the bounds improve significantly, though the authors note that for certain $\theta$ , the unconditional bounds are surprisingly close to those assuming DH.

Significance and Claims
The paper claims to provide the "most explicit relation" between zero-density estimates and exceptional set bounds to date. Its significance lies in:

Unifying Framework: It creates a systematic method to translate any improvement in zero-density estimates (either $A(\sigma)$ or $A^*(\sigma)$ ) directly into improved bounds on the exceptional set for the PNT in short intervals.
Utilization of Additive Energy: By explicitly incorporating the additive energy of zeros ( $A^*(\sigma)$ ), the authors demonstrate that fourth-moment estimates are necessary to obtain the sharpest known unconditional bounds, recovering and refining results that relied on older, less precise density estimates.
Computational Verification: The authors emphasize that their theoretical bounds are immediately applicable via computer assistance, allowing for the direct application of the latest results (e.g., Guth–Maynard) without requiring new analytic machinery for the exceptional set itself.

The authors remain modest, noting that the exact formulae for the bounds are complex and likely to be superseded as new zero-density estimates are proven. They do not claim to resolve the PNT in short intervals for $\theta \le 2/15$ but rather provide the best known quantitative description of the exceptions in the currently accessible range.