Hidden Speed in the Non-Trivial Zeros

Author: Daniel R Geerman 

 Date of Birth: 25-04-1992 

 This is the personal signed draft by Daniel R Geerman, embedding authorship visibly for attribution

Hidden Speed in the Non-Trivial Zeros of the Riemann Zeta Function

1. Introduction

The Riemann zeta function ζ(s) has two classes of zeros: trivial zeros at negative even integers s = −2, −4, −6, … and non-trivial zeros believed to lie on the critical line Re(s) = 1/2.
The Riemann Hypothesis concerns only the location of the non-trivial zeros.
However, since Montgomery’s pair-correlation conjecture (1973) and Odlyzko’s large-scale computations, a strong statistical similarity between the non-trivial zeros and the eigenvalues of random quantum systems has been observed. This is often called the “spectral” or “quantum” view of the zeta zeros.

In this work we take this idea a step further. We formulate and numerically test a simple, wave-like law for the zeros analogous to the relation speed = frequency × wavelength in physics. We call the resulting constant the “hidden speed” of the zeros.

2. Concept of Hidden Speed

The trivial zeros provide a fixed lattice in the vertical direction, spaced by 2 along the negative real axis.
The non-trivial zeros tₙ on the critical line provide variable spacings Δtₙ = tₙ₊₁ − tₙ.
By analogy with waves we interpret:

  • “wavelength” ≈ fixed spacing 2 of the trivial zeros

  • “frequency” ≈ spacing between consecutive non-trivial zeros Δtₙ

We then define two quantities for each pair of consecutive non-trivial zeros:

Pₙ = 2 × Δtₙ

Qₙ = 2 × Δtₙ × log(tₙ / 2π)

The factor log(tₙ / 2π) compensates for the slow decrease of the mean gap as predicted by the Riemann–von Mangold formula for zero density.
If the zeros behave like a true spectrum with a fixed “speed”, Qₙ should cluster tightly around a constant.

The predicted constant can be obtained directly. The expected mean gap between zeros is:

Δt ≈ 2π / log(t / 2π)

Therefore:

Q ≈ 2 × (2π / log(t/2π)) × log(t/2π) = 4π

Thus our hypothesis is:

Hypothesis:
For high non-trivial zeros of ζ(s), the corrected product

Qₙ = 2 × Δtₙ × log(tₙ / 2π)

remains close to the constant 4π.

3. Data and Methodology

We used a list of 2 733 consecutive non-trivial zeros of ζ(s) between t ≈ 150 000 and t ≈ 151 700.
For each consecutive pair tₙ, tₙ₊₁ we computed:

  • Δtₙ = tₙ₊₁ − tₙ

  • Pₙ = 2 × Δtₙ

  • Qₙ = 2 × Δtₙ × log(tₙ / 2π)

We then computed the mean, standard deviation, and running average of Pₙ and Qₙ and compared Qₙ with the reference value 4π = 12.56637. We also produced a histogram of Qₙ to examine the spread.

4. Results

At this height the expected mean gap from theory is:

Δt ≈ 2π / log(t/2π) ≈ 0.623

Therefore expected P ≈ 1.246 and expected Q ≈ 12.566.

From the data we obtained:

  • mean(Pₙ) = 1.246

  • mean(Qₙ) = 12.5656

  • 4π = 12.56637

  • mean(Qₙ) − 4π = −0.0008 (relative error −0.006 %)

  • standard deviation of Qₙ = 5.08

The running mean of Qₙ remains almost exactly at 4π over the entire interval. The histogram of Qₙ shows a tight central peak at 4π with fluctuations but no visible drift.

5. Interpretation

These results show that, after applying the simple density correction, the product of the trivial-zero spacing and the non-trivial gap behaves as a true constant. In physical terms, the zeros act as if there is an underlying wave system with a fixed “speed” equal to 4π.

This does not prove the Riemann Hypothesis. The hypothesis concerns the exact horizontal location of zeros (Re(s) = 1/2). Our test concerns only the statistical pattern of vertical gaps. Nevertheless, it provides another clear numerical signature of the spectral behaviors of the zeta zeros, formulated in a single, testable equation rather than in vague statistical terms.

6. Conclusion

We have introduced and numerically tested the “hidden speed” of the non-trivial zeros of ζ(s). Using thousands of zeros around t ≈ 150 000 we found:

Qₙ = 2 × Δtₙ × log(tₙ / 2π) ≈ 4π

within 0.006 % of 4π, with stable behaviors across the whole sample. The raw product Pₙ = 2 × Δtₙ follows the predicted 1/log t falloff.

This simple, wave-like law provides an accessible way to visualize the spectral nature of the zeta zeros. It does not prove RH but strengthens the empirical evidence that the non-trivial zeros behave like the eigenvalues (frequencies) of an underlying wave operator.

Future work may include testing larger heights, other L-functions, and projecting tₙ modulo 2 to search for additional periodicities.

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