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Extreme values of the real part of the Riemann zeta function

This paper extends the work of Bondarenko and Seip by establishing new lower bounds for the extreme values of the real part of the Riemann zeta function on the critical line, utilizing the resonance method and integral estimates for Dirichlet series with non-negative coefficients.

Original authors: Qiyu Yang, Shengbo Zhao

Published 2026-07-07

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Qiyu Yang, Shengbo Zhao

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 Riemann zeta function, $\zeta(s)$ , as a giant, invisible musical instrument that plays a continuous, complex melody across the universe of numbers. Mathematicians have long been fascinated by the "loudest" notes this instrument can play. These loud notes correspond to the function reaching its extreme values (very large numbers) at specific points on a special line called the "critical line."

For decades, researchers have tried to figure out just how loud these notes can get.

The Old Way: Listening to the Volume

Previously, mathematicians like Soundararajan and later Bondarenko and Seip developed a clever technique called the "resonance method." Think of this like a sound engineer trying to find the loudest possible volume of the instrument.

To do this, they built a special "resonator"—a mathematical tool designed to vibrate in sync with the zeta function. When the resonator and the zeta function vibrate together, they amplify the signal. By measuring this amplified signal, they could prove that the zeta function gets very loud.

However, there was a catch. The zeta function's output is a complex number, which has two parts: a real part (like the volume) and an imaginary part (like a hidden phase or direction). The old methods measured the total volume (the modulus, $|\zeta|$ ). This told them the note was loud, but it didn't tell them which way the sound wave was pointing. It could be loud because the real part was huge, or because the imaginary part was huge, or a mix of both.

The New Discovery: Tuning the Direction

In this paper, authors Qiyu Yang and Shengbo Zhao take the existing resonance method and tweak it to listen specifically to the real part of the sound.

Imagine the zeta function's output as an arrow pointing in a specific direction on a map.

Old Method: Measured the length of the arrow (how far it is from the center).
New Method: Measures how far the arrow points specifically to the East (the real part).

The authors' main achievement (Theorem 1.1) is proving that the zeta function doesn't just get loud in a random direction; it gets extremely loud specifically in the "East" direction (the real part).

How They Did It: The "Constructive Interference" Trick

The secret sauce in their method is a mathematical trick involving convolution.

The Problem: The zeta function is a mix of positive and negative numbers, which can cancel each other out (like noise-canceling headphones).
The Solution: The authors use their resonator to mix the zeta function with a special filter. This filter acts like a prism that separates the light.
The Result: After passing through this filter, the "noise" (negative parts) is minimized, and the "signal" (positive parts) is amplified. It's like turning a chaotic crowd of people shouting in different directions into a single choir singing in perfect harmony.

Because the resulting signal is made of non-negative coefficients (all positive numbers), the "East" direction (the real part) gets a massive boost. The authors prove that this boost is so strong that the real part of the zeta function reaches a record-breaking height, almost as high as the total volume itself.

Why This Matters

The paper shows that the zeta function is not just "loud" in a general sense; it is directionally loud.

Before: We knew the zeta function could reach a peak height of $H$ .
Now: We know that at that same peak height, the "East" component is also nearly $H$ .

This is a significant refinement. It tells us that the extreme values aren't just random spikes in the complex plane; they are robust, positive surges.

The Corollaries (Side Discoveries)

The authors also show that this "directional loudness" applies to other related mathematical objects:

Derivatives: Even if you look at the rate of change of the zeta function (how fast the melody is speeding up or slowing down), the real part still hits these massive peaks.
Near the Line: Even if you step slightly off the "critical line" (the main stage), the real part remains incredibly loud, though slightly less so than right on the line.

Summary

In simple terms, Yang and Zhao took a powerful tool used to measure the total volume of a mathematical sound and refined it to measure the volume in a specific direction. They proved that the Riemann zeta function doesn't just get loud; it gets positively, directionally loud, shattering previous records for how high its real part can climb. This confirms that the "extreme values" of this famous function are not just chaotic fluctuations but have a strong, positive structure.

Technical Summary: Extreme Values of the Real Part of the Riemann Zeta Function

Problem Statement
The paper addresses the problem of establishing lower bounds for the extreme values of the real part of the Riemann zeta function, $\text{Re}(\zeta(1/2 + it))$ , on the critical line. While significant progress has been made in understanding the extreme values of the modulus $|\zeta(1/2 + it)|$ , the behavior of the real part specifically remains a distinct challenge. Existing results for the modulus do not provide phase information; a large modulus does not guarantee that the real part is large and positive, as the magnitude could be dominated by the imaginary part. The authors aim to demonstrate that $\zeta(1/2 + it)$ can achieve large values specifically in the direction of the real axis.

Methodology
The authors employ the resonance method, originally introduced by Soundararajan and subsequently refined by Bondarenko and Seip. The core of their approach involves the following steps:

Resonator Construction: Following the framework of Bondarenko and Seip, the authors construct a specific resonator $R(t) = \sum_{m \in M'} r(m)m^{-it}$ . This involves defining a set of primes $P$ and a multiplicative function $f(n)$ supported on square-free numbers. The set $M'$ is carefully constructed to maximize the contribution of the Dirichlet polynomial associated with the resonator.
Convolution Formula: A key technical innovation in this work is the use of a convolution formula to transform $\zeta(1/2 + it)$ into a Dirichlet polynomial with non-negative coefficients. By integrating $\text{Re}(\zeta(1/2 + i(t+u)))$ against a kernel $K(u)$ and the squared modulus of the resonator $|R(t)|^2$ , the authors isolate the real part.
Integral Estimation: The proof relies on estimating the integral $\int_{\mathbb{R}} \text{Re}(\zeta(1/2 + it)) |R(t)|^2 \Phi(t/T) dt$ $\int_{R} Re (ζ (1/2 + i t)) ∣ R (t) ∣^{2} Φ (t / T) d t$ , where $\Phi$ $Φ$ is a Gaussian weight function.
- Lower Bound: Using Lemma 2.2, the authors establish a lower bound for the integral involving Dirichlet series with non-negative coefficients. The positivity of the coefficients after convolution is interpreted as a form of constructive interference, amplifying the positive real contribution without requiring the imaginary part to be small.
- Upper Bounds: The authors utilize classical convexity bounds and moment estimates for $\zeta(s)$ to bound the error terms arising from the tails of the integrals and the regions where the convolution is not dominant.

Key Contributions and Results
The primary result is Theorem 1.1, which establishes a new lower bound for the maximum of the real part of the zeta function on the critical line over the interval $[T^\beta, T]$ .

Theorem 1.1: For a fixed $\beta \in [0, 1)$ and sufficiently large $T$ :
$\max_{t \in [T^\beta, T]} \text{Re} \zeta\left(\frac{1}{2} + it\right) \ge \exp\left( \left(\sqrt{1-\beta} + o(1)\right) \sqrt{\frac{\log T \log_3 T}{\log_2 T}} \right)$
where $\log_k$ denotes the $k$ -th iterated logarithm.
Corollary 1: The method is extended to the real parts of the derivatives of the zeta function, $\text{Re}(\zeta^{(\ell)}(1/2 + it))$ , yielding the same order of magnitude for the lower bound.
Corollary 2: The results are extended to points near the critical line, specifically $\sigma = 1/2 + A/\log_2 T$ . The lower bound remains of the same order, though the constant in the exponent is slightly reduced to $\lambda(A) = \frac{1}{e^A\sqrt{e}-1}$ .

Significance and Claims
The paper claims that its results provide a directional refinement of the known extreme values of $\zeta(1/2 + it)$ .

Simultaneous Extremality: The authors emphasize that their result implies the simultaneous existence of extreme values for both the modulus and the real part. Unlike previous methods that might force the imaginary part to be small to maximize the real part, this approach relies on the convolution formula transforming the zeta function into a polynomial with non-negative coefficients, thereby amplifying the real contribution through constructive interference.
Relation to Existing Work: The authors note that their result immediately implies the lower bounds for the modulus established by Bondarenko and Seip, as $|\zeta(s)| \ge \text{Re}(\zeta(s))$ .
Limitations: The paper modestly acknowledges that improving the constant in the exponent from $(\sqrt{1-\beta} + o(1))$ to $(\sqrt{2} + o(1))$ —a bound achieved for the modulus by de la Bretèche and Tenenbaum—appears difficult using the current method. The authors explain that a large lower bound for $|\zeta(1/2 + it)|^2$ does not guarantee a large positive real part, as the magnitude could be driven by the imaginary component. Thus, the specific technique used here is distinct from the double-version convolution used to optimize the modulus bounds.

In summary, the paper successfully adapts the resonance method to isolate and amplify the real part of the Riemann zeta function, proving that $\zeta(1/2 + it)$ can attain large values in a prescribed direction (the real axis) with a specific asymptotic growth rate.