What question did this study set out to answer?

To numerically verify Mokrane's Exact Law regarding the spectral coefficients of the Riemann zeta function.

May 6, 20260 citationsOpen Access

Mokrane's Exact Law: Numerical Verification of the Precise Asymptotic 2|cₙ|γₙ = 2 - 1/ (4γₙ²) + O (γₙ^-4)

Key Points

To numerically verify Mokrane's Exact Law regarding the spectral coefficients of the Riemann zeta function.
Verified Mokrane's Exact Law to better than 10^{-9} precision over 2 million zeros.
Derived a sharp bound for |ψ(p_K) - p_K| with confirmation through numerical analysis.
Utilized data from the Odlyzko dataset to compute zeros and apply discrete projection methods.
All 2,000,000 computed zeros satisfy |Re(ρ_n) - 1/2| < 10^{-12}.
The sharp bound was confirmed with explicit constant C = 1/π.

Abstract

This archive contains the complete numerical data, source code, and documentation for the discovery of Mokrane's Exact Law governing the spectral coefficients of the Riemann zeta function. MAIN SCIENTIFIC FINDINGS: 1. Mokrane's Exact Law: 2|cₙ|γₙ = 2 - 1/ (4γₙ²) + O (γₙ^-4) verified to better than 10^-9 precision over 2, 000, 000 zeros. 2. Sharp bound: |ψ (pK) -pK|/pK^3/2 = log (pK) / (π pK) + O (1/pK) with explicit constant C = 1/π derived theoretically and confirmed numerically. 3. Numerical evidence: All 2, 000, 000 computed zeros satisfy |Re (ρₙ) -1/2| < 10^-12. DATA INCLUDES: - First 2, 000, 000 zeros (γₙ) from Odlyzko dataset- Spectral coefficients |cₙ| computed via discrete projection- All numerical results in CSV format- Complete Python source code with Numba parallelization- High-resolution figures (PNG format) RELATED WORK: An earlier version of this work, focusing on spectral stability, has been submitted to the International Journal of Number Theory (IJNT). Author: Ahmed MokraneORCID: 0009-0005-2319-6098Date: 2026-05-04

AI से पूछें

Bookmark

View Full Paper