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A discrete mean-value theorem for the higher derivatives of the Riemann zeta function

The paper establishes that the sum of the $n$ th derivative of the Riemann zeta function over its non-trivial zeros is real and exhibits a specific sign pattern in the mean (positive for odd $n$ , negative for even $n$ ) by deriving a full asymptotic expansion of these sums.

Original authors: Christopher Hughes, Andrew Pearce-Crump

Published 2026-05-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Christopher Hughes, Andrew Pearce-Crump

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: Listening to the "Heartbeat" of Numbers

Imagine the Riemann Zeta function ( $\zeta(s)$ ) as a massive, complex machine that governs the distribution of prime numbers (like 2, 3, 5, 7, 11...). This machine has a special set of "hidden gears" called non-trivial zeros. These are specific points where the machine's output hits exactly zero.

For a long time, mathematicians have been fascinated by what happens when you look at the "speed" or "acceleration" of this machine right at those hidden gears. In math terms, this means taking the derivative (the rate of change) of the function at these zero points.

Shanks' Conjecture was an old idea (from the 1970s) suggesting that if you look at the first derivative (the speed) at all these zeros and add them up, the result is a real, positive number. It's like saying if you listen to the "heartbeat" of the machine at every zero, the average beat is always positive.

What This Paper Does: The "Higher-Order" Upgrade

Christopher Hughes and Andrew Pearce-Crump decided to take this idea and crank it up to the next level. Instead of just looking at the speed (1st derivative), they looked at the acceleration (2nd derivative), jerk (3rd derivative), and so on, all the way up to the $n$ -th derivative.

Think of it like this:

The Old View: You only checked the car's speedometer at specific checkpoints.
This Paper: They checked the speedometer, the acceleration, the jerk, and every other motion sensor at those same checkpoints, and then added up all those readings.

The Main Discovery: A Pattern of Signs

The authors discovered a beautiful, predictable pattern in the "average" result of these sums:

If you look at an odd derivative (1st, 3rd, 5th...), the average sum is positive.
If you look at an even derivative (2nd, 4th, 6th...), the average sum is negative.

It's like a pendulum swinging. If you measure the swing at odd intervals, it's always on the right; at even intervals, it's always on the left. The paper proves this isn't just a lucky guess; it's a fundamental rule of how the Zeta function behaves.

How They Did It: The "Residue" Detective Work

To prove this, the authors didn't just guess; they used a sophisticated mathematical toolkit involving complex analysis (a branch of math dealing with imaginary numbers).

The Contour Map: Imagine drawing a giant rectangular fence around a field of zeros.
The Integral: They calculated a "flow" (an integral) along the edges of this fence. In math, a famous rule (Cauchy's Theorem) says that if you know the flow around the fence, you know exactly what's happening inside the fence (the sum of the derivatives at the zeros).
The "Left" Side: They found that the flow along three sides of the fence was negligible (like a gentle breeze). The real "storm" of data was coming from the left side of the fence.
The Expansion: They broke down the complex formulas on that left side into a long list of terms (an asymptotic expansion). This is like taking a complex recipe and listing every single ingredient and its exact amount, rather than just saying "add some flour."

The Result: A Precise Recipe

The paper provides a full recipe (an asymptotic formula) for calculating these sums.

The Main Ingredient: The biggest part of the answer depends on how high you go up the number line (represented by $T$ ) and involves logarithms (a way of measuring growth).
The Seasoning: They also calculated the smaller, "lower-order" terms (the $C_j$ and $A_j$ constants mentioned in the text). These are like the pinch of salt or dash of pepper that makes the prediction accurate, not just a rough estimate.
The Error Margin: They also calculated exactly how much their prediction might be off (the error term). They showed that even without assuming the famous "Riemann Hypothesis" (a huge unsolved math problem), their prediction is incredibly close. If you do assume the Riemann Hypothesis is true, the prediction becomes even sharper.

Why It Matters (According to the Paper)

The paper doesn't claim to solve the Riemann Hypothesis or apply this to physics or engineering. Instead, its value is purely mathematical:

Generalization: It takes a known fact about the 1st derivative (Shanks' Conjecture) and proves it works for all derivatives.
Precision: Previous studies only calculated the "main term" (the biggest part of the answer). This paper calculates the entire expansion, including the smaller, messy details.
Verification: They tested their formula for the 2nd derivative ( $n=2$ ) against actual computer data for the first 100,000 zeros. The math matched the data almost perfectly, with the error being tiny (less than 2,400 on a total value of millions).

Summary Analogy

Imagine you are trying to predict the total noise level of a choir singing a complex song.

Shanks figured out that if you listen to the volume of the singers at specific notes, the average is positive.
Hughes and Pearce-Crump figured out that if you listen to the change in volume, the change in that change, and so on, there is a strict rule: odd changes are positive, even changes are negative.
They didn't just guess the volume; they wrote down a precise mathematical formula that tells you exactly what the volume should be, including all the tiny nuances, and proved it works by comparing it to a recording of the actual choir.

Technical Summary: A Discrete Mean-Value Theorem for the Higher Derivatives of the Riemann Zeta Function

Problem Statement
The paper addresses the problem of determining the asymptotic behavior of the discrete mean value of the $n$ -th derivative of the Riemann zeta function, $\zeta^{(n)}(s)$ , evaluated at the non-trivial zeros $\rho = \beta + i\gamma$ of $\zeta(s)$ . Specifically, the authors seek to evaluate the sum:
$\sum_{0 < \gamma \leq T} \zeta^{(n)}(\rho)$
While the case $n=1$ (Shanks' Conjecture) was previously established by Conrey, Ghosh, Gonek, and Fujii, the behavior for higher derivatives ( $n \geq 2$ ) remained less explored. The authors aim to generalize these results to all $n$ , providing explicit asymptotic formulae that include lower-order terms and precise error bounds, both unconditionally and under the assumption of the Riemann Hypothesis (RH).

Methodology
The proof follows a contour integration approach, utilizing Cauchy's Residue Theorem. The methodology proceeds through the following steps:

Contour Integral Setup: The authors define a rectangular contour $R$ with vertices $c+i, c+iT, 1-c+iT, 1-c+i$ (where $c = 1 + 1/\log T$ ). By Cauchy's Theorem, the integral of $\frac{\zeta'(s)}{\zeta(s)}\zeta^{(n)}(s)$ along this contour equals the sum of residues at the non-trivial zeros, which is the target sum $\sum \zeta^{(n)}(\rho)$ .
Contour Decomposition: The integral is split into four segments: bottom ( $S_B$ ), top ( $S_T$ ), right ( $S_R$ ), and left ( $S_L$ ). The contributions from $S_B$ , $S_T$ , and $S_R$ are shown to be negligible (contained within the error term) using convexity bounds for $\zeta(s)$ and its derivatives, and bounds for the logarithmic derivative $\frac{\zeta'}{\zeta}(s)$ .
Evaluation of the Left-Hand Side: The main contribution comes from the left-hand side of the contour ( $S_L$ ). Using the functional equation of the zeta function and its derivatives (specifically Lemma 4 and Lemma 3), the integral is transformed into an expression involving Dirichlet series.
Reduction to Arithmetic Sums: Through the application of Gonek's Lemma 5, the integral is reduced to a discrete arithmetic sum involving the von Mangoldt function $\Lambda(r)$ :
$(-1)^{n+1} \sum_{mr \leq T/2\pi} \Lambda(r) \log^n r$
Perron's Formula and Residue Calculation: This arithmetic sum is evaluated using a truncated Perron's formula, converting it back into a complex integral. The integral is then evaluated by shifting the contour to the left, picking up the residue at the pole $s=1$ .
Laurent Expansion: The residue at $s=1$ is calculated by expanding the integrand $\left(\frac{\zeta'(s)}{\zeta(s)}\right)^{(n)} \zeta(s) \frac{Y^s}{s}$ (where $Y = T/2\pi$ ) using Laurent series expansions around $s=1$ . This involves coefficients $C_j$ from the expansion of $\zeta(s)$ and $A_j$ from the expansion of $\frac{\zeta'(s)}{\zeta(s)}$ .
Error Term Analysis: The error terms are analyzed in two regimes:
- Unconditionally: By choosing a specific height $V$ for the contour shift, the error is bounded by $O(T e^{-C\sqrt{\log T}})$ .
- Conditionally (RH): Assuming the Riemann Hypothesis, the error is improved to $O(T^{1/2} \log^{n+5/2} T)$ . A specific improvement for $n=1$ is also derived, yielding $O(T^{1/2} \log^{13/4} T)$ .

Key Contributions and Results
The primary contribution is Theorem 3, which provides a full asymptotic expansion for the sum of the $n$ -th derivatives at the zeros:

$\sum_{0<\gamma \leq T} \zeta^{(n)}(\rho) = (-1)^{n+1} \frac{1}{n+1} \frac{T}{2\pi} \log^{n+1}\left(\frac{T}{2\pi}\right) + \text{Lower Order Terms} + E_n(T)$

Key features of the result include:

Generalization: It extends Shanks' Conjecture (the $n=1$ case) to all $n \geq 1$ .
Explicit Lower Order Terms: Unlike previous works (e.g., Kaptan, Karabulut, and Yıldırım) which only calculated the main term, this paper provides explicit coefficients for all lower-order terms involving powers of $\log(T/2\pi)$ and constants $C_j$ and $A_j$ .
Sign of the Mean: The results imply that $\zeta^{(n)}(\rho)$ is real and positive in the mean if $n$ is odd, and real and negative in the mean if $n$ is even.
Numerical Verification: The authors provide numerical evidence for the $n=2$ case, showing that the asymptotic formula agrees with the true sum of the first 100,000 zeros with high precision (error never exceeding 2,400 for a sum magnitude of $\approx 10^6$ ).

Significance
The paper claims to provide the first complete asymptotic expansion for the mean value of higher derivatives of the zeta function at its zeros. By recovering the known results for $n=1$ (Fujii's theorem) and extending them to $n=2$ and beyond with explicit lower-order terms, the work offers a more refined understanding of the distribution of these derivatives. The authors note that their method allows for the calculation of any specific $n$ -th order derivative, though higher orders become computationally unwieldy by hand. The work serves as a rigorous generalization of Shanks' Conjecture, confirming the alternating sign pattern of the mean values based on the parity of the derivative order.