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Joint extreme values of $L$ -functions on and off the critical line

This paper unconditionally demonstrates that any number of distinct primitive $\mathrm{GL}(1)$ and $\mathrm{GL}(2)$ $L$ -functions can simultaneously attain large values on the critical line, while also analyzing their joint distribution to the right of the critical line using the resonance method and a novel variation of Heath-Brown's technique to avoid reliance on zero-distribution information.

Original authors: Athanasios Sourmelidis

Published 2026-05-06

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Athanasios Sourmelidis

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 the world of mathematics as a vast, mysterious ocean. In this ocean, there are special waves called L-functions. These aren't just any waves; they are mathematical structures that encode deep secrets about numbers, much like how the tides encode secrets about the moon.

For over a century, mathematicians have been trying to understand the "extreme values" of these waves—specifically, how high they can spike. The paper you are asking about, written by Athanasios Sourmelidis, is a breakthrough in understanding what happens when multiple of these waves crash into each other at the same time.

Here is the story of the paper, broken down into simple concepts:

1. The Goal: A Synchronized High-Five

Imagine you have a group of surfers (the L-functions). Each surfer has their own wave. Sometimes, one surfer catches a massive, record-breaking wave. That's interesting. But the big question is: Can all the surfers catch a massive wave at the exact same moment?

Mathematicians call this "joint extreme values." If they can, it means the universe of numbers has a moment of synchronized chaos where everything is huge at once.

2. The Old Rules (The "Critical Line")

There is a specific "lane" in the ocean called the Critical Line (mathematically, the line where the real part of the number is 1/2). This is the most important lane.

The Previous Problem: Before this paper, a team of mathematicians (Heap and Li) proved that if you have 2 or 3 surfers, they can definitely catch a big wave together. But if you wanted to prove this for 4 or more surfers, they had to make a huge assumption: they had to believe that all the waves in the ocean follow a perfect, predictable pattern (a hypothesis called the Riemann Hypothesis). Without this assumption, the math broke down.
The New Breakthrough: Sourmelidis says, "We don't need to assume the waves are perfect to prove they can crash together." He developed a new method to show that any number of these waves (GL(1) and GL(2) types) can simultaneously reach huge heights on the critical line. He did this without needing the "perfect pattern" assumption. It's like proving the surfers can high-five without needing to know exactly how the wind is blowing.

3. The New Tools: The "Resonance" Method

How did he do it? He used a technique called the Resonance Method.

The Analogy: Imagine you have a tuning fork. If you hit it, it vibrates at a specific frequency. If you have a second tuning fork nearby, it might start vibrating too, even if you didn't hit it, because it "resonates" with the first one.
In Math: Sourmelidis built a special mathematical "tuning fork" (a Dirichlet polynomial). He tuned it so that it vibrates in perfect harmony with the specific L-functions he was studying. When this "tuning fork" is applied, it amplifies the waves, making it possible to prove they can get very large.

4. The "Off-Line" Adventure

The paper also looks at what happens off the critical line (in the deeper, less explored parts of the ocean).

Here, the waves are a bit wilder. To prove they can get big together, Sourmelidis had to assume that the "dead zones" (places where the waves are zero) aren't too crowded.
He showed that even with this mild assumption, the waves can still synchronize and reach extreme heights. This improves upon previous work by other mathematicians who had to make stronger, more restrictive assumptions.

5. The "Fractional Moment" Trick

One of the most clever parts of the paper is a variation of a method invented by Heath-Brown.

The Analogy: Imagine you want to measure the average height of a crowd, but you can only look at them through a blurry lens. Usually, you need a very clear picture (perfect knowledge of where the zeros are) to get an accurate average.
The Trick: Sourmelidis found a way to use a "fractional" lens. Instead of needing a perfect picture, he used a mathematical trick that allowed him to get a good enough average even with the blurry lens. This is what allowed him to avoid needing the "perfect pattern" assumption for the critical line.

Summary

In short, this paper is a victory for unconditional proof.

Before: "We can prove these numbers get huge together, but only if we assume a very strict rule about how they behave."
Now: "We can prove these numbers get huge together, no matter what, using a new resonance technique and a clever mathematical trick."

Sourmelidis has shown that the chaotic, extreme behavior of these number-theoretic waves is a robust feature of the universe, not just a fluke that happens under perfect conditions. He didn't just fix a small error; he removed a massive crutch that mathematicians had been leaning on for decades.

Technical Summary: Joint Extreme Values of L-Functions on and Off the Critical Line

Problem Statement
The paper addresses the problem of establishing the existence of simultaneous large values (extreme values) for a tuple of distinct primitive $L$ -functions. Specifically, it investigates whether $H$ distinct primitive $L$ -functions from the classes $GL(1)$ and $GL(2)$ can simultaneously attain large magnitudes at the same point $t$ on the critical line ( $\sigma = 1/2$ ). Previous work by Heap and Li established such results for $H=2$ and $H=3$ unconditionally, but for $H > 3$ , their results relied on the assumption of the Generalized Riemann Hypothesis (GRH). Furthermore, while results existed for $L$ -functions off the critical line (where $\sigma > 1/2$ ), they often required strong assumptions or produced values at distinct points $t_h$ rather than a single simultaneous $t$ . The paper aims to provide unconditional results for joint extreme values on the critical line for any number of $GL(1)$ and $GL(2)$ $L$ -functions and to extend these findings to the region off the critical line under zero-density estimates.

Methodology
The primary analytical tool employed is the resonance method, originally introduced by Soundararajan. The core strategy involves constructing a "resonator" Dirichlet polynomial $R(s)$ designed to amplify the values of the target $L$ -functions.

On the Critical Line ( $\sigma = 1/2$ ):
- The proof relies on establishing a lower bound for a twisted moment integral involving the product of the $L$ -functions and the resonator, and an upper bound for the same integral twisted by the product of the individual $L$ -functions.
- A key innovation is the adaptation of Heath-Brown's method for fractional moments of the Riemann zeta-function. Instead of assuming the GRH to control the zeros of the $L$ -functions, the author utilizes a variation of Heath-Brown's technique to obtain sharp lower bounds for twisted fractional moments.
- This approach allows the author to avoid assuming information about the zero distribution of $L$ -functions with degree less than three. The method involves splitting the $L$ -function product into a short Dirichlet polynomial approximation and an error term, then applying Gabriel's convexity theorem to control the difference.
- The proof utilizes Hölder's inequality to decouple the joint moment into individual moments, leveraging known sharp upper bounds for the second moments of $GL(1)$ and $GL(2)$ $L$ -functions.
Off the Critical Line ( $\sigma > 1/2$ ):
- The methodology shifts to utilizing zero-density estimates. The author identifies "zero-free" horizontal strips where the $L$ -functions do not vanish.
- Within these strips, the $L$ -functions can be approximated by short Dirichlet polynomials without the need for the GRH, provided the zero-density estimates grow sufficiently slowly (specifically $o(T^{1-\alpha})$ ).
- The proof employs a resonator constructed by Hilberdink and Voronin.
- For Theorem 1.3, which deals with non-simultaneous values (distinct $t_h$ ), the paper utilizes a convolution formula involving the Fejér kernel and the quantitative Kronecker theorem. A technical refinement allows the application of Kronecker's theorem directly to disjoint sets of primes for each $L$ -function, thereby bypassing the need for the orthogonality of Dirichlet coefficients (condition 1.1) in the specific construction of the simultaneous points.

Key Contributions and Results

Theorem 1.1 (Unconditional Joint Extremes on the Critical Line):
- The paper proves that for any $H$ distinct primitive $L$ -functions from the class $S_0$ (containing the Riemann zeta-function, primitive Dirichlet $L$ -functions, and $GL(2)$ cusp forms), there exists a $t \in [T, 2T]$ such that:
  $\max_{t \in [T, 2T]} \min_{h \le H} |L_h(1/2 + it)| \ge \exp\left( \sqrt{\frac{D \log T}{H(H-1) \log \log T}} \right)$
- This result is unconditional for any $H$ , removing the GRH assumption required by Heap and Li for $H > 3$ .
- The constant $D$ is admissible for $D < 1/2$ if all functions are Dirichlet $L$ -functions. For general $GL(2)$ forms, $D < \frac{1/2 - \theta}{3 + \theta}$ , where $\theta$ is the bound towards the Ramanujan Hypothesis.
- The bound is slightly weaker than the conditional Heap-Li bound by a factor of $1/\sqrt{H-1}$ , but it is the first unconditional result for arbitrary $H$ .
Theorem 1.2 (Joint Extremes Off the Critical Line):
- Under the assumption of the Ramanujan Hypothesis and specific zero-density estimates ( $\sum N_{L_h}(\sigma_0, T) = o(T^{1-\alpha})$ ), the paper establishes that for $\sigma \in (\sigma_0, 1)$ , there exists a $t \in [T, 2T]$ such that the $L$ -functions can simultaneously attain large values (or small values for the reciprocal) of size $\exp\left( D (\log T)^{1-\sigma} / \log \log T \right)$ .
- This result unconditionally covers $GL(2)$ holomorphic cusp forms and improves upon previous work by Mahatab, Pańkowski, and Vatwani by ensuring the values occur at a single $t$ rather than a collection of nearby $t_h$ .
Theorem 1.3 (Non-Simultaneous Extremes for General Classes):
- For a broader class $S$ of $L$ -functions (without requiring the Ramanujan Hypothesis or polynomial Euler products), the paper proves the existence of $t_1, \dots, t_H$ (where $|t_h - t_g|$ is small) such that the $L$ -functions attain large values.
- This generalizes results by Inoue and Li and recovers their findings for Dirichlet $L$ -functions under weaker assumptions.

Significance and Claims
The paper claims to be an unconditional improvement of the general result by Heap and Li regarding joint extreme values on the critical line. By introducing a variation of Heath-Brown's method for fractional moments, the author successfully circumvents the need for the Generalized Riemann Hypothesis when dealing with more than three $L$ -functions.

The author notes that while the lower bound in Theorem 1.1 is slightly smaller than the conditional bound (due to the $H(H-1)$ term in the denominator versus $H$ ), the removal of the GRH assumption is the primary achievement. The paper also highlights that the restriction on the constant $D$ and the specific power of $H$ in the bound are direct consequences of the limitations in controlling the twisted fractional moments unconditionally.

Regarding the off-critical line results, the paper claims to partially recover results by Inoue and Li and generally improve upon the work of Mahatab et al. by demonstrating that simultaneous large values can be found at a single point $t$ under zero-density estimates, without requiring the full strength of the GRH. The work is presented as a rigorous extension of the resonance method to tuples of $L$ -functions, bridging the gap between single-function extreme value theory and joint distribution theory.