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New large value estimates for Dirichlet polynomials

This paper establishes new bounds on the frequency of large values for Dirichlet polynomials near the critical size of $N^{3/4}$ , leading to an improved zero density estimate for the Riemann zeta function and asymptotic results for primes in short intervals of length $x^{17/30+o(1)}$ .

Original authors: Larry Guth, James Maynard

Published 2026-04-09

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CC BY 4.0

Original authors: Larry Guth, James Maynard

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 are trying to listen to a specific radio station (the "signal") in a room filled with static noise. In the world of mathematics, specifically Analytic Number Theory, the "signal" is often a pattern hidden within prime numbers, and the "noise" is the chaotic behavior of complex mathematical waves called Dirichlet polynomials.

For decades, mathematicians have been trying to answer a simple question: How often can these mathematical waves get unexpectedly loud?

If a wave gets too loud (takes a "large value"), it can mess up our ability to predict where prime numbers are hiding. The louder the wave, the harder it is to find the primes.

Larry Guth and James Maynard have just published a breakthrough paper that acts like a new, super-sensitive noise-canceling headset. They have found a way to prove that these waves cannot get as loud as we previously thought, at least not as often as we feared.

Here is the breakdown of their discovery using everyday analogies:

1. The Problem: The "Loud Party"

Think of a Dirichlet polynomial as a party where thousands of guests (numbers) are shouting at once.

The Length ( $N$ ): The number of guests.
The Volume ( $V$ ): How loud the party is.
The Goal: We want to know how many times the party gets so loud that it drowns out everything else.

Previously, mathematicians had a rule of thumb (the "Mean Value Theorem") that said: "If the party has $N$ guests, it might get loud enough to be heard clearly about $N$ times."
However, there was a "danger zone" (around a volume of $N^{3/4}$ ) where the old rules were too loose. It was like saying, "This party might be loud 100 times," when in reality, it was probably only loud 50 times. That 50% difference matters a lot when you are trying to solve the Riemann Hypothesis (the ultimate puzzle of prime numbers).

2. The Solution: The "Additive Energy" Detective

Guth and Maynard didn't just look at the volume; they looked at the dance floor.

They introduced a concept called Additive Energy. Imagine the guests at the party.

Low Energy: The guests are scattered randomly. If you pick four people, it's unlikely that two pairs are standing in a perfect square formation.
High Energy: The guests are organized. Maybe they are standing in a grid, or a line, or a perfect circle. If you pick four people, they often form neat patterns.

The Old Way: Mathematicians assumed the worst-case scenario: the guests were perfectly organized (High Energy), making the party as loud as possible.
The New Way: Guth and Maynard realized that in the "danger zone," the guests cannot be perfectly organized. They proved that if the party gets very loud, the guests must be somewhat chaotic.

They used a clever trick involving matrices (grids of numbers) and singular values (a way to measure the "strength" of a grid). They showed that for the party to be loud, the "dance floor" (the set of points where the wave is loud) has to be very messy. But if it's messy, the wave actually cancels itself out and gets quieter!

3. The "Affine Transformation" Analogy

One of their key insights involves a concept called Affine Transformations.
Imagine you have a small group of people standing in a specific shape (a triangle). Now, imagine you have a machine that can stretch, shrink, or rotate that shape.

The Question: Can you find a small group of people that stays a triangle no matter how many times you stretch or rotate them?
The Answer: No. If you keep stretching and rotating, the shape eventually breaks apart.

Guth and Maynard proved that the "loud spots" of the mathematical wave cannot stay in a neat, organized shape under these mathematical "stretching" operations. Because the loud spots can't hold their shape, the wave can't stay loud for long. This allows them to lower the estimate of how often the wave gets loud.

4. The Results: Why Should We Care?

This isn't just about abstract math; it has real consequences for how we understand Prime Numbers.

The Zero Density Estimate: The Riemann Zeta function is a giant machine that generates prime numbers. It has "zeros" (places where the machine stops working). We know these zeros exist, but we want to know how many are in the "danger zone" (where they mess up our predictions).
- Old Estimate: "There might be a lot of zeros here."
- New Estimate: "There are significantly fewer zeros here."
- The Math: They improved the exponent from roughly 2.4 to 2.3 (specifically $30/13$ ). In the world of prime numbers, shaving off a tiny fraction of an exponent is like finding a new continent.
Primes in Short Intervals: This helps us answer: "If I look at a short stretch of numbers (like between 1,000,000 and 1,000,100), how many primes will I find?"
- Old Result: We could only guarantee finding primes in intervals of length $x^{7/12}$ (about 58% of the number size).
- New Result: We can now guarantee it in intervals of length $x^{17/30}$ (about 56.6% of the number size).
- Why it matters: It means we can find primes in shorter intervals than ever before. It's like being able to find a needle in a haystack that is 10% smaller than before.

Summary

Larry Guth and James Maynard took a problem that had been stuck for decades—estimating how loud certain mathematical waves get—and solved it by realizing that loud waves require organized patterns, but these specific waves are too chaotic to stay organized.

By proving that the "dance floor" of these waves is messy, they proved the waves must be quieter than we thought. This quieting effect ripples out to give us sharper, more accurate tools for counting prime numbers, bringing us one step closer to cracking the code of the universe's most famous numbers.

1. Problem Statement

The central problem addressed is the frequency of large values of Dirichlet polynomials. Specifically, given a Dirichlet polynomial $D(t) = \sum_{N < n \le 2N} b_n n^{it}$ with bounded coefficients ( $|b_n| \le 1$ ), the authors seek to bound the number of points $R$ in a $1$-separated set $W \subset [0, T]$ where $|D(t)| \ge V$ .

This problem is critical in analytic number theory because the behavior of the Riemann zeta function $\zeta(s)$ and the distribution of prime numbers are intimately linked to the large values of such polynomials. The "critical regime" for many applications occurs when the length of the polynomial is $N$ and the value threshold is $V \approx N^{3/4}$ .

Previous State of the Art:
Prior to this work, the best bounds were derived from:

The Mean Value Theorem: Effective for small $V$ .
The Montgomery-Halász-Huxley (MHH) estimate: Effective for large $V$ .
The combination of these two methods (Equation 1.1 in the paper) provided a bound of roughly $R \ll T^{o(1)}(N^2 V^{-2} + T \min(NV^{-2}, N^4 V^{-6}))$ . However, in the critical range $N^{7/10} \lesssim V \lesssim N^{8/10}$ (specifically near $V=N^{3/4}$ ), these previous bounds were not optimal.

2. Methodology

The authors introduce a novel method that combines linear algebra (singular values), harmonic analysis (Poisson summation and stationary phase), and additive combinatorics (additive energy).

A. Matrix Formulation and Singular Values

The problem is reformulated using a matrix $M_W$ of size $|W| \times N$ with entries $(M_W)_{t,n} = w(n/N)n^{it}$ , where $w$ is a smooth weight function. The condition $|D(t)| \ge N^\sigma$ implies a lower bound on the operator norm (largest singular value $s_1(M_W)$ ).
Using the trace of powers of the matrix $M_W M_W^*$ , specifically the cubic trace $\text{tr}((M_W M_W^*)^3)$ , the authors relate the number of large values to the singular values. They utilize a refined inequality (Lemma 4.2) to bound the largest singular value by the difference between the cubic trace and the cube of the trace, effectively isolating the contribution of the "non-trivial" terms.

B. Decomposition of the Trace

The cubic trace expansion leads to a sum $S$ over triples of points in $W$ and integers. This sum is decomposed into three parts based on the frequency variables $m_1, m_2, m_3$ arising from Poisson summation:

$S_1$ (One non-zero frequency): Shown to be negligible via rapid decay of Fourier transforms.
$S_2$ (Two non-zero frequencies): Controlled using Heath-Brown's Theorem (Theorem 1.6), which bounds sums over difference sets. This part relies on the additive structure of $W$ .
$S_3$ (Three non-zero frequencies): This is the most difficult term and the core of the new contribution.

C. Analysis of $S_3$ and Additive Energy

The term $S_3$ involves an integral over variables $v_1, v_2$ and a sum over frequencies $m$ . The integrand contains a function $R(v) = \sum_{t \in W} |v|^{it}$ .

Cancellation Mechanism: The authors observe that the inner integral over the third variable $u_3$ exhibits massive cancellation unless the linear combination $m_1 v_1 + m_2 v_2 + m_3 \approx 0$ . This restricts the domain of integration to a thin neighborhood of a curve.
Additive Energy ( $E(W)$ ): The size of $S_3$ $S_{3}$ is controlled by the additive energy of the set $W$ $W$ , defined as $E(W) = \#\{(t_1, t_2, t_3, t_4) \in W^4 : |t_1+t_2-t_3-t_4| \le 1\}$ $E (W) = # {(t_{1}, t_{2}, t_{3}, t_{4}) \in W^{4} : ∣ t_{1} + t_{2} - t_{3} - t_{4} ∣ \leq 1}$ .
- If $E(W)$ is small (random-like sets), standard square-root cancellation arguments yield strong bounds.
- If $E(W)$ is large (structured sets), the authors use a refined argument involving affine transformations. They prove that a set $W$ cannot be "approximately closed" under a large family of affine maps $u \mapsto (m_1 u + m_3)/m_2$ unless it has very specific structure, which is ruled out by Heath-Brown's bounds.

This leads to a new bound for $S_3$ that interpolates between the random case and the structured case, improving upon previous estimates when $E(W)$ is large but not maximal.

3. Key Contributions

New Large Value Estimate (Theorem 1.1):
The authors prove a new bound for the number of large values $R$ :
$R \le T^{o(1)} \left( N^2 V^{-2} + N^{18/5} V^{-4} + T N^{12/5} V^{-4} \right)$
This improves upon the classical MHH bound specifically in the range $N^{7/10+\epsilon} < V < N^{8/10-\epsilon}$ .
Refined Handling of Additive Energy:
The paper provides a sophisticated analysis of how the additive energy of the set of large values affects the bound. They show that even for sets with high additive energy (which usually leads to worse bounds), the specific structure of Dirichlet polynomials prevents the worst-case scenarios from occurring, provided the energy is not maximal.
Technical Innovation in $S_3$ :
The derivation of the bound for the cubic term $S_3$ using the interplay between the geometry of the integration domain (the curve $m_1 v_1 + m_2 v_2 + m_3 = 0$ ) and the Fourier decay of the smoothed sum $R(v)$ is a significant technical breakthrough.

4. Main Results and Applications

A. Zero Density Estimate for the Riemann Zeta Function

The primary application is an improved bound on the number of zeros $N(\sigma, T)$ of $\zeta(s)$ with real part $\ge \sigma$ and imaginary part $\le T$ .

New Bound: $N(\sigma, T) \le T^{\frac{30(1-\sigma)}{13} + o(1)}$ .
Improvement: This improves the previous exponent of $12/5 = 2.4$ (from Huxley) to $30/13 \approx 2.307$ .
Significance: This is a quantitative improvement in the "density hypothesis" regime, particularly for $\sigma$ close to $3/4$ .

B. Distribution of Primes in Short Intervals

The improved zero density estimate translates directly into better results for the distribution of primes in short intervals $[x, x+y]$ .

Corollary 1.3 (Pointwise): For $y \in [x^{17/30+\epsilon}, x^{0.99}]$ $y \in [x^{17/30 + ϵ}, x^{0.99}]$ , the prime count satisfies the asymptotic $\pi(x+y) - \pi(x) \sim y/\log x$ $π (x + y) - π (x) \sim y / lo g x$ .
- Improvement: The exponent $17/30 \approx 0.566$ improves on Huxley's $7/12 \approx 0.583$ .
Corollary 1.4 (Almost All): For $y \in [X^{2/15+\epsilon}, X^{0.99}]$ $y \in [X^{2/15 + ϵ}, X^{0.99}]$ , the asymptotic holds for all but a negligible set of $x$ $x$ .
- Improvement: The exponent $2/15 \approx 0.133$ improves on Huxley's $1/6 \approx 0.166$ .

5. Significance

This paper represents a substantial advance in the analytic theory of the Riemann zeta function. By breaking the barrier of the classical Montgomery-Halász-Huxley bound in the critical range $V \approx N^{3/4}$ , Guth and Maynard have:

Sharpened the Zero Density Theorem: Providing the best known unconditional bound for the density of zeros in the critical strip for a significant range of $\sigma$ .
Extended Short Interval Results: Pushing the known range of validity for the prime number theorem in short intervals closer to the conjectured limit ( $y \approx x^{1/2}$ ).
Introduced New Techniques: The method of combining singular value analysis with additive energy bounds and affine transformation arguments offers a new toolkit for tackling similar problems in analytic number theory, potentially applicable to other L-functions or exponential sums.

The result is particularly notable because the improvement is not just a minor logarithmic gain but a significant reduction in the exponent, achieved by fundamentally re-evaluating the structure of large value sets for Dirichlet polynomials.