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A Gaussian-Perron Prime-Side Defect and Local Profiles Near Critical-Line Zeros of the Riemann Zeta Function

This paper introduces a Gaussian-Perron prime-force defect that transforms the explicit formula into a local diagnostic for the geometry of Riemann zeta zeros, proving that under the Riemann Hypothesis and specific damping conditions, the defect near critical-line zeros exhibits a universal selected-zero profile with exponentially small nonlocal contributions.

Original authors: Netzer Moriya

Published 2026-07-09

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Netzer Moriya

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 as a giant, invisible musical instrument stretched across the universe. For over a century, mathematicians have been trying to tune it, specifically looking for a hidden pattern in its "notes" (called zeros) that might prove the Riemann Hypothesis. Usually, they try to listen to the whole orchestra at once, which is incredibly noisy and confusing.

This paper, by Netzer Moriya, introduces a clever new way to listen: instead of trying to hear the whole symphony, they build a super-sensitive, laser-focused microphone that zooms in on just one single note at a time.

The "Force" and the "Defect"

Think of the Zeta function as a landscape with hills and valleys. The mathematicians are interested in the "slope" of this landscape right next to a specific zero (a spot where the music stops). They call this slope the "horizontal force."

Usually, you can calculate this force by adding up the contributions of all the prime numbers (the building blocks of arithmetic) or by looking at all the zeros. But doing both at once is messy.

The author creates a new tool called the Gaussian–Perron prime-force defect. Imagine this as a special filter.

The Prime Side: It takes the chaotic noise of all the prime numbers and smooths it out using a "Gaussian" curve (a bell shape) and a "Perron" cutoff (a sharp stop). This turns the infinite, messy list of primes into a neat, manageable signal.
The Defect: The paper then measures the difference (the "defect") between this smoothed prime signal and the actual mathematical force of the Zeta function.

The Magic of Zooming In

Here is the cool part: When you zoom in very close to a specific zero (let's call it $\rho_0$ ), using a special magnifying glass scale where the distance is measured as $1/\log X$ , something magical happens.

The paper proves that if you look at this "defect" right next to a zero that sits exactly on the "critical line" (the middle of the strip where the Riemann Hypothesis says all the zeros should be), the messy noise from all the other zeros and primes disappears. It gets suppressed so effectively that it becomes almost invisible.

What's left is a universal profile. It's like a fingerprint that is exactly the same for every single zero on that line. The paper shows that this fingerprint follows a specific, clean mathematical shape: $-a \text{ Re}(e^{-\lambda}/\lambda)$ .

What About Zeros That Don't Belong?

The paper explicitly rules out a different scenario. It asks: "What if there is a zero that is not on the critical line?" (This would mean the Riemann Hypothesis is false).

The authors show that if such a "rogue" zero existed, the signal wouldn't look like a neat, stable fingerprint. Instead, the "defect" would grow linearly and wildly as you zoom in. It would look like a spike that keeps getting bigger and bigger, rather than settling into a calm pattern. The paper doesn't find these rogue spikes; it just says, "If they were there, they would look totally different."

How Sure Are They?

The authors are very confident about the math they have done. They have proved that:

The smoothed prime signal creates an exact error-function weight (a specific bell-curve shape).
If you assume the Riemann Hypothesis is true (that all zeros are on the line) and that the "smoothing width" is chosen correctly, the noise from other zeros is suppressed exponentially. This means the "defect" near a zero is almost entirely determined by that one zero itself.
The formula for this local shape is exact, with only tiny, exponentially small errors remaining.

They also simulated this idea on a computer. They picked the very first non-trivial zero of the Zeta function (located at $1/2 + 14.1347251417347 i$ ) and ran a calculation using 2,000,000 prime numbers. The result? The computer's "measured" defect matched the theoretical prediction almost perfectly, with a difference so small it was only $3.40 \times 10^{-10}$ . This isn't a proof that the Riemann Hypothesis is true, but it's a very strong, high-precision check that their new "microscope" works exactly as the math says it should.

The Bottom Line

This paper doesn't solve the Riemann Hypothesis. Instead, it builds a new, local diagnostic tool. It shows that if you look at the Zeta function through this specific Gaussian-smoothed lens, the chaotic global noise vanishes, leaving behind a clean, predictable, and universal shape for every zero on the critical line. It's like turning a static-filled radio into a crystal-clear channel, but only when you tune it to the exact frequency of a "good" zero. If the radio starts screaming with linear noise, it might mean a "bad" zero is hiding nearby, but the paper doesn't find any of those.

Technical Summary: A Gaussian–Perron Prime-Side Defect and Local Profiles Near Critical-Line Zeros

Problem Statement
The paper addresses the local geometry of the non-trivial zeros of the Riemann zeta function, $\zeta(s)$ , specifically in the vicinity of a selected zero $\rho_0$ . While the Riemann Hypothesis (RH) and related horizontal monotonicity properties concern the global behavior of the completed logarithmic derivative $\xi'/\xi$ , this work focuses on constructing a local diagnostic tool. The objective is to isolate the contribution of a single zero from the "cloud" of other zeros and prime-power contributions using a smoothed prime-side force. The central question is whether a smoothed prime-field approximation can be constructed such that the defect between this field and the actual logarithmic derivative $\zeta'/\zeta$ reveals a universal local profile near a critical-line zero, with all non-selected contributions exponentially suppressed.

Methodology
The author introduces a Gaussian–Perron prime-force defect, denoted $\Delta_{X,\alpha}(s)$ , defined as:
$\Delta_{X,\alpha}(s) = \left(\sigma - \frac{1}{2}\right) \text{Re}\left( P_{X,\alpha}(s) - \frac{\zeta'}{\zeta}(s) \right)$
where $P_{X,\alpha}(s)$ is a Gaussian-smoothed prime field constructed via a contour integral involving the kernel $H_{X,\alpha}(z) = X^z \exp(\alpha^2 Y z^2)$ with $Y = \log X$ .

The methodology proceeds through the following analytical steps:

Prime-Side Representation: The smoothed field $P_{X,\alpha}(s)$ is shown to correspond to a weighted sum over prime powers, where the weights are given by the complementary error function, $WX,\alpha(n) = \frac{1}{2}\text{erfc}\left(\frac{\log n - \log X}{2\alpha\sqrt{Y}}\right)$ . This provides a rapidly decaying cutoff centered at $n=X$ .
Contour Shifting and Explicit Formula: By shifting the contour of the integral representation of $P_{X,\alpha}(s)$ $P_{X, α} (s)$ from $\text{Re}(z) = c$ $Re (z) = c$ to $\text{Re}(z) = -d$ $Re (z) = - d$ , the author derives an explicit formula for the defect. This formula decomposes the defect into contributions from:
- Non-trivial zeros ( $\rho$ ) within the strip $-d < \text{Re}(\rho-s) < c$ .
- The pole at $s=1$ .
- Trivial zeros at negative even integers.
- A shifted contour integral remainder.
Local Scaling: The analysis is conducted in the logarithmic local scale $s = \rho_0 + \lambda / \log X$ , where $\rho_0$ is a fixed simple zero and $\lambda = a + ib$ is a complex parameter.
Anisotropic Damping: A key analytical tool is the Gaussian damping functional $Q_\alpha(\rho'; \rho_0) = \text{Re}(\rho' - \rho_0) + \alpha^2 \text{Re}((\rho' - \rho_0)^2)$ . This functional determines the exponential decay rate of non-selected zero contributions. The paper establishes that vertical displacements from $\rho_0$ are damped, while horizontal displacements can increase weight, necessitating specific geometric conditions on the zero distribution.

Key Contributions and Results

Exact Selected-Zero Profile: The paper derives an exact formula for the contribution of a single selected zero $\rho_0$ to the defect.
- Critical-Line Case ( $\rho_0 = 1/2 + i\gamma_0$ ): Under the assumption that $\rho_0$ is simple and on the critical line, the selected zero contribution converges to a universal bounded profile:
  $L_{\rho_0, X, \alpha}(s) \sim -a \text{Re}\left(\frac{e^{-\lambda}}{\lambda}\right)$
  where $a = \text{Re}(\lambda)$ . This profile is independent of the smoothing parameter $\alpha$ in the leading order.
- Off-Line Case ( $\beta_0 \neq 1/2$ ): For a hypothetical zero off the critical line, the contribution scales linearly with $\log X$ , specifically $O(\log X)$ . This provides a theoretical distinction between critical and off-critical zeros within this defect framework.
Localization Theorem: The main result (Theorem 7.6) establishes that, under the Riemann Hypothesis and specific damping conditions, the full defect $\Delta_{X,\alpha}(s)$ near a critical-line zero is dominated by the selected-zero profile, with the nonlocal remainder (contributions from other zeros, the pole, and the contour) being exponentially small ( $O(e^{-cY})$ ).
- Conditions: The theorem requires a "pole-damping condition" (a lower bound on $\alpha$ relative to $\gamma_0$ ) and "contour regularity" (bounds on $\zeta'/\zeta$ on the shifted line).
- RH Implication: Assuming RH, the zero-spacing condition is automatically satisfied because all non-trivial zeros lie on the line $\text{Re}(s)=1/2$ , ensuring $Q_\alpha < 0$ for all $\rho' \neq \rho_0$ .
Numerical Verification: Section 8 provides a direct numerical check of the prime-side defect using a truncated prime-power sum. For the first non-trivial zero, the computed defect matches the theoretical finite- $X$ selected-zero profile with high precision (maximum discrepancy $\approx 3.4 \times 10^{-10}$ ), validating the analytic derivation in a computable regime.

Significance and Claims
The paper claims to provide a local diagnostic for zero geometry associated with the Riemann zeta function. Unlike previous works that study the global monotonicity of $|\xi|$ or $\text{Re}(\xi'/\xi)$ , this work isolates the local behavior of the "horizontal force" near a specific zero.

The significance lies in:

Decoupling: Successfully separating the contribution of a single zero from the global sum of zeros and primes using a Gaussian–Perron kernel.
Universality: Demonstrating that the local profile near a critical-line zero is universal (independent of $\alpha$ and the specific zero's height, provided it is simple).
Diagnostic Potential: The framework suggests that the defect $\Delta_{X,\alpha}$ could serve as a tool to detect deviations from the critical line (off-line zeros would exhibit linear growth in $\log X$ rather than a bounded profile), contingent on the ability to control the nonlocal remainder.

The authors remain modest regarding applications, stating that the analysis is "local and defect-theoretic" and focused on the asymptotic behavior of the defect within the RH-level setting. The work does not propose new algorithms for computing zeros or prove the Riemann Hypothesis; rather, it constructs a theoretical framework to analyze the local structure of the zeta function's zeros should they exist.