The Dual-Zero Lattice Function

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


The Dual-Zero Lattice: A New Geometric Lens on the Riemann Zeta Function

In my “Law of Imbalance” posts I argued that tiny asymmetries can create large-scale structures. Here I apply that philosophy to the zeros of the Riemann zeta function and arrive at a new geometric representation: the Dual-Zero Lattice (DZL).

1. Motivation

When mathematicians study the non-trivial zeros of the zeta function, the trivial zeros are usually ignored. But if we take them seriously, they provide a natural scaffolding for a completely different picture of the zero distribution — one that makes the density law and fluctuations visible at a glance.

2. Building the lattice

  • Vertical lines: at the trivial zeros x = −2n (or shifted so that the critical line Re(s) = ½ is one edge).

  • Horizontal lines: at the non-trivial zero ordinates y = γₖ on Re(s) = ½.

The intersections form rectangular cells whose:

  • width = spacing of trivial zeros (W = 2),

  • height = gap between consecutive non-trivial zeros Δγₖ = γₖ₊₁ − γₖ.

I call this grid of cells the Dual-Zero Lattice (DZL).

3. Cells and triangles

Each cell represents one “zero as a whole.” Draw a diagonal from the trivial-zero side to the critical line and the cell splits into two congruent triangles. Geometrically:

  • Cell area: Aₖ = W * Δγₖ.

  • Triangle area: Aₖ^Δ = Δγₖ.

  • Diagonal angle: θₖ = arctan(Δγₖ / W).

This simple picture encodes the unity/duality theme of my imbalance idea: the cell embodies unity; the two triangles embody duality within that unity.

4. Quantitative invariants

Combine these shapes with the known density of zeros and three clean invariants emerge:

  • Cell scaling: Iₖ = Aₖ * log(γₖ / 2π) → 4π.

  • Triangle scaling: Jₖ = Aₖ^Δ * log(γₖ / 2π) → 2π.

  • Angle form: Tₖ = tan(θₖ) * log(γₖ / 2π) → π.

The first two correspond to the Riemann–von Mangold law. The third — the angle form — drops straight out of the DZL construction and is new as a visual/diagnostic expression.



5. What the data show

I computed these invariants for published tables of non-trivial zeros at both low and high heights (up to 10⁵+). In all cases:

  • Iₖ oscillates tightly around 4π with no drift.

  • Jₖ oscillates around 2π with no drift.

  • Tₖ oscillates around π with no drift.

Running means stay flat; residuals are
centered. This confirms that the DZL is a faithful geometric encoding of the zero statistics and a simple diagnostic for anomalies.

6. Significance

Even though the invariants themselves reflect known theory, the construction is new. By combining trivial and non-trivial zeros, the DZL:

  • makes the density law immediately visible as shrinking cell height,

  • offers an intuitive “unity/duality” visual metaphor,

  • gives simple, easy-to-compute diagnostics (areas, angles) for testing zero distribution at any height.

This geometric picture provides a fresh way of “seeing” the Riemann Hypothesis and may inspire further questions about higher-order structure and correlations.

7. Next steps

Future work on the DZL could explore:

  • autocorrelation of the residuals Tₖ − π to test clustering,

  • extreme-value behavior of cell and triangle areas,

  • comparison with random-matrix models using the angle form,

  • extensions to other L-functions.

Skai’s note: This is part of my ongoing exploration of imbalance and symmetry in mathematics. The Dual-Zero Lattice offers a new way to visualize the interplay of trivial and non-trivial zeros — a lens I haven’t seen elsewhere. Even when it confirms known results, the act of “seeing” them differently is itself a form of discovery.







Comments

Post a Comment

Popular posts from this blog

The Law of Imbalance

Image
 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 The Law of Imbalance: From Ramanujan’s Primes to the Structure of the Universe Abstract  I propose that the difference between Ramanujan’s prime-counting approximation and the true distribution of primes is not an error but a universal principle. This remainder, ∆(x) = π(x) − Li(x), reflects the same asymmetry observed in physics, where matter slightly outweighed antimatter in the early universe, allowing stars, planets, and life to exist. If the Riemann Hypothesis is true, then the imbalance in primes is forever bounded but never gone, just as physical asymmetry is eternal. I argue that the “law of imbalance” is the hidden structure of reality: conservation requires action and reaction, but existence requires imbalance. Hypothesis (Law of Imbalance): For E(x)=Δ(x)/(√x·ln x) there exist constants ...
Read more

A Perimeter Law for Local Zero Geometry (DZL)

 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 A Perimeter Law for Local Zero Geometry (DZL) and a Universal + π 2 / log ⁡ +\pi^2/\log + π 2 / log Correction Abstract We introduce a simple geometric construction on three consecutive unfolded gaps between zeros (the “DZL triangle/perimeter”). Across all tests we ran—Riemann zeta zeros (your list at height ∼ 10 5 \sim10^5 ∼ 1 0 5 ), random-matrix surrogates (CUE), and an L L L -function surrogate—the same pattern emerges: Area/angle lanes: three linear combinations of local gaps converge (locally, in moving windows) to the constants I → 4 π , J → 2 π , T → π \boxed{I\to 4\pi,\quad J\to 2\pi,\quad T\to \pi} I → 4 π , J → 2 π , T → π ​ . Perimeter second-order: defining K k △    =    ( s k − 1 + s k )    +    s k − 1 2 + s k 2 , Z k    =    ( K k △ − 2 π )   Λ k , K_k^\triangle \;=\; (s_{k-1}+s_k)\...
Read more