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Contributions to the theory of the Euler eta function

This paper investigates the distribution of the zeros of Euler's eta function along with its integral and fractional derivatives.

Original authors: Torre Caparatta, Sebastian Pauli, Filip Saidak

Published 2026-08-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Torre Caparatta, Sebastian Pauli, Filip Saidak

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

In the vast landscape of mathematics, there are certain numbers that act as keys to understanding the hidden order of the universe. Among these, the most famous are the prime numbers, the indivisible building blocks of arithmetic. For centuries, mathematicians have tried to find a pattern in how these primes are scattered, a quest that led to the discovery of a mysterious function known as the Riemann zeta function. This function behaves like a complex map, where the most important locations are its "zeros"—specific points where the function's value drops to nothing. The arrangement of these zeros is believed to hold the secret to the distribution of prime numbers, a connection so profound that proving it is one of the greatest unsolved challenges in science. While the original map is difficult to study directly, mathematicians often look at related maps, or variations, to gain new perspectives. One such variation is the Euler eta function, a slightly different version of the original that avoids a specific mathematical singularity, making it a smoother, more manageable object to examine. By studying how this smoother map behaves, researchers hope to uncover clues about the more chaotic original.

A team of mathematicians from the University of North Carolina at Greensboro has recently taken a deep dive into this smoother map, exploring not just the zeros of the function itself, but also the zeros of its derivatives. In simple terms, a derivative measures how a function changes at any given point, and by looking at these changes, the researchers could trace the paths that zeros take as the function evolves. They investigated what happens when they take these derivatives, including fractional ones, which are a way of measuring change that falls between whole-number steps. Their work reveals a fascinating and somewhat surprising landscape. They found that for most of the complex plane, the zeros of these derivatives follow predictable paths, often staying within specific vertical strips or moving along clear lines. They were able to prove that in certain wide regions, particularly on the right side of the map, the function never hits zero, effectively creating safe zones where the value is guaranteed to be non-zero.

However, the story becomes more intricate on the left side of the map. Here, the researchers discovered that the zeros of the first derivative are confined to the real number line, meaning they do not wander into the complex, imaginary realm in that specific area. They proved that these real zeros appear in a regular pattern, with exactly one zero located between every pair of negative even integers. This finding adds a layer of order to a region that might otherwise seem chaotic. Furthermore, the team mapped out how the zeros of the original function connect to the zeros of its derivatives. They traced continuous paths that link a zero of the original function to a zero of a derivative, showing how these points migrate as the mathematical operation changes. These paths act like bridges, connecting different parts of the mathematical structure and revealing how the zeros of the original function are intimately tied to the behavior of its derivatives.

One of the most striking discoveries in this research is the existence of a double zero, a rare event where two paths of zeros meet and merge into a single point. This occurs at a specific location on the negative real axis, near a value of minus four point eight eight, and happens when the derivative is taken to a fractional power of approximately zero point four seven. This double zero is unique; the researchers note that it is the only multiple zero of a fractional derivative they have found for this function or for the related Riemann zeta function. While the existence of this point is confirmed through rigorous calculation and simulation, the team admits they do not yet have a complete theoretical explanation for why it happens. It stands as a singular anomaly in an otherwise orderly system, a small but significant mystery that challenges their current understanding.

The researchers also looked at the total number of zeros in these functions as they move further out along the imaginary axis. They found that the number of zeros for the derivatives of the Euler eta function grows at the same rate as the number of zeros for the original Riemann zeta function. This suggests a deep, underlying similarity between the two, despite their different starting points. They proposed that this relationship holds true even for fractional derivatives, suggesting that the fundamental structure of these zeros is preserved regardless of how the function is changed. While they have proven many of these patterns, they also identified open questions, particularly regarding whether any non-real zeros exist for the derivatives on the left side of the map. Their work provides a clearer picture of the terrain, confirming that while the Euler eta function shares many properties with its more famous cousin, it also possesses its own unique quirks, such as the mysterious double zero, that continue to invite further exploration.

Problem Statement
This paper investigates the distribution of zeros of the Euler eta function, $\eta(s)$ (also known as the alternating zeta or Dirichlet eta function), and its derivatives, including both integer and fractional orders. While the zeros of the Riemann zeta function $\zeta(s)$ and its derivatives are well-studied, the authors aim to establish parallel and distinct properties for $\eta(s)$ . The primary objectives are to identify zero-free regions for $\eta(s) - c$ and its derivatives $\eta^{(\alpha)}(s)$ , determine the non-vanishing of the first derivative $\eta'(s)$ in the left half of the critical strip under the assumption of the Riemann Hypothesis (RH) for sufficiently large imaginary parts, locate zeros of $\eta'(s)$ on the negative real axis, and derive asymptotic estimates for the zero-counting functions $N_k^\eta(T)$ .

Methodology
The authors employ a combination of analytic number theory techniques, complex analysis, and numerical computation:

Series Dominance and Rouché's Theorem: To locate zeros in the right half-plane, the authors analyze the Dirichlet series representation of the derivatives. They identify regions where a single term $Q_M^\alpha(s) = (\log M)^\alpha / M^s$ dominates the rest of the series. By applying Rouché's theorem to the difference between the full derivative and the dominant terms, they prove the existence and simplicity of zeros in specific rectangular regions.
Functional Equations and Logarithmic Derivatives: The study of zeros in the critical strip and the left half-plane relies heavily on the functional equation of $\eta(s)$ and its relationship to $\zeta(s)$ . The authors utilize logarithmic derivatives and the argument principle to count zeros and establish bounds on the real parts of zeros.
Inequalities and Asymptotic Analysis: The proofs for zero-free regions and zero locations involve detailed estimation of trigonometric and gamma function terms, often using bounds derived from the Riemann Hypothesis or known properties of the Gamma function (e.g., Stirling's formula).
Numerical Verification: Theoretical findings are supported by numerical computations performed using the Sage computer algebra system and the mpmath multi-precision library. These computations visualize paths of zeros connecting $\eta(s)$ to its derivatives and derivatives to $\eta(s)-c$ .

Key Contributions and Results

Zero-Free Regions for $\eta(s) - c$ :
- The authors prove that $\eta(s) \neq 1$ for $\sigma > \sigma_0 \approx 1.9401$ , where $\sigma_0$ is the zero of a specific function involving $\zeta(\sigma)$ .
- They establish conditions involving $|\sin(t \log 2)|$ and $-\cos(t \log 2)$ that define zero-free regions for $\eta(s) - c$ where $c \in [0, 1)$ .
- They observe that paths of zeros connecting $\eta(s)$ to $\eta(s)-1$ behave differently depending on their starting point: paths starting at zeros of $\zeta(s)$ lead to zeros of $\eta(s)-1$ , while paths starting at the non-trivial zeros of $\eta(s)$ (specifically $s_n = 1 + 2n\pi i / \log 2$ ) have real parts that approach infinity as $c \to 1$ .
Zero-Free Regions and Zeros of Derivatives in the Right Half-Plane:
- The paper generalizes results from $\zeta^{(k)}(s)$ to $\eta^{(\alpha)}(s)$ for real $\alpha > 0$ .
- Theorem 3.1 establishes explicit zero-free regions for $\eta^{(\alpha)}(s)$ based on the dominance of specific terms in the Dirichlet series.
- Theorem 3.2 proves that in specific vertical strips $S_M^\alpha$ , there is exactly one simple zero of $\eta^{(\alpha)}(s)$ for each integer $j$ , located near the line where two consecutive terms of the series cancel each other out.
- Unlike $\zeta^{(\alpha)}(s)$ , which has no zeros on the positive real axis, the alternating nature of $\eta(s)$ allows for zeros of $\eta^{(\alpha)}(s)$ on the real axis.
Non-Vanishing of $\eta'(s)$ in the Critical Strip (assuming RH):
- Theorem 4.1 demonstrates that if the Riemann Hypothesis is true, then $\eta'(s) \neq 0$ for $0 < \sigma < 1/2$ and $|t| \ge 164$ . This is derived by analyzing the real part of the logarithmic derivative $-\eta'/\eta(1-s)$ and showing it is positive under the RH assumption for sufficiently large $|t|$ .
Zeros of $\eta'(s)$ in the Left Half-Plane:
- Theorem 5.1 and Theorem 5.4 locate the zeros of $\eta'(s)$ on the negative real axis. They prove that for $n \ge 3$ , there exists a zero in the interval $[-2n-2, -2n-2+\epsilon]$ .
- The authors show that $\eta'(s)$ has only real zeros for $\sigma \le 0$ and exactly one real zero in each interval $(-2n-2, -2n)$ .
Zero-Counting Functions:
- Theorem 6.1 establishes that the number of zeros of the $k$ -th derivative $\eta^{(k)}(s)$ with imaginary part up to $T$ , denoted $N_k^\eta(T)$ , satisfies $N_k^\eta(T) = N_\zeta(T) + O(\log T)$ as $T \to \infty$ . This parallels the known result for $\zeta^{(k)}(s)$ and suggests a one-to-one correspondence between the zeros of $\zeta$ and the derivatives of $\eta$ via the paths of zeros of $\eta(s)-c$ .

Significance and Observations
The paper highlights both parallels and surprising differences between the theory of $\zeta(s)$ and $\eta(s)$ . While many results regarding zero distributions are analogous, the authors note a unique phenomenon: the existence of a double zero for a fractional derivative.

Numerical evidence (visualized in Figures 4 and 7) reveals a double zero of $\eta^{(\alpha)}(s)$ at $\alpha \approx 0.473$ near $s \approx -4.88$ . This occurs where paths of zeros from the left half-plane meet the real axis. The authors note this is the only multiple zero of a fractional derivative discovered for these functions so far, a feature not observed in the standard theory of $\zeta(s)$ or Dirichlet L-functions.
The authors propose Conjecture 7.1, extending the zero-counting asymptotic to fractional derivatives ( $N_\eta^\alpha(T) = N_\zeta(T) + O(\log T)$ ).
They also propose Conjecture 7.2 (attributed to Ricky Farr), stating that for integer $k$ , $\eta^{(k)}(s)$ has no non-real zeros in the left half-plane ( $\sigma < 0, t \neq 0$ ), with the exception of the specific double zero phenomenon observed for fractional derivatives.

The work serves to complement the classical theory of the Riemann zeta function by mapping the specific landscape of the Euler eta function, providing rigorous bounds for its derivatives, and identifying unique structural features like the double zero that distinguish it from its non-alternating counterpart.

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