Hello Wolfram Community,
I'm exploring the spectral properties of a custom-built operator inspired by the zeros of the Riemann zeta function, and I wanted to share my idea and get feedback or suggestions on it. Background & Motivation:The non-trivial zeros of the Riemann zeta function are deeply connected to the distribution of prime numbers. Many researchers have approached this through quantum chaos, random matrix theory, and spectral analysis of various operators. Inspired by these ideas, I designed a toy operator that combines discretized derivatives, potential terms, and other energy contributions, aiming to mimic some spectral features related to the zeros. My Operator Construction:
I discretized a domain (logarithmic or uniform grid) with about 800 points.
The operator includes several terms:
A kinetic term with a scaled second derivative (Laplacian).
A potential term based on prime numbers, involving cosine functions of their logs.
Geometric terms like Ricci curvature, modeled as diagonal matrices.
Additional energy contributions such as fluid-like movements, bubble tensions, eye functions, Planet Jupiter, brain neurological functions, heart frequency and functions and cosmic terms, scaled appropriately. Please have a look at my picture files I attached with this post and you'll see the results.
And go to my zenodo profile to download my codes so you can run these your selfs along with other information there too. Here it is: https://doi.org/10.5281/zenodo.19199926
I assemble these into a Hermitian matrix (operator) in Mathematica/Wolfram Language, ensuring symmetry.
Key Challenge & Goal:
Initially, the spectrum of this operator doesn't reach the eigenvalues corresponding to the 100th zero (~241.7) of the zeta function.
To analyze the spectral statistics and compare eigenvalue distributions to those of the zeros, I need the spectrum to encompass at least up to that range.
I am exploring scaling parameters (like lapscale, Vprime_scale, etc.) to stretch the spectrum.
My current approach involves automatically increasing these scales until the largest eigenvalue surpasses that threshold.
What I’m Asking:
Do you have suggestions on alternative ways to scale or modify my operator to reach higher spectral ranges?
Are there known tricks or modifications in spectral theory or quantum chaos models that could help?
Would adding a constant shift or other transformations affect the spectral unfolding in a beneficial way?
Any ideas on improving the construction or ensuring the eigenvalues' distribution better mimics the zeros?
Additional details:
I included visualizations: raw eigenvalue spectrum, unfolded levels, and level spacing distributions.
I compare my eigenvalues to the known zeros of the zeta function, and I measure the RMS difference after unfolding and scaling.
Links & Files:
I can share the full code (Mathematica or Python) if interested, for reproduction or further experimentation.
Thanks in advance for your insights, suggestions, or discussion!
Attachments:
