Asymmetry  for the Riemann  Hypothesis

Walid OUKIL

doi:10.33774/coe-2025-97vw5-v16

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Asymmetry for the Riemann Hypothesis

30 June 2026, Version 16

Working Paper

Walid OUKIL

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This content is an early or alternative research output and has not been peer-reviewed by Cambridge University Press at the time of posting.

Abstract

We show that, under a certain rigidity inequality, the Riemann zeta function satisfies $(\zeta(s),\zeta(1-\overline{s}))\neq(0,0)$ for every $s$ in the critical strip off the critical line. The verification of this rigidity inequality is left to number theorists.

Keywords

Zeta function

Riemann Hypothesis

Euler differential equation

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Here is a numerical test of the rigidity inequality from Theorem applied to the Riemann zeta function: https://colab.research.google.com/drive/1ICP-fBZgE9t3elHp9X4B5BqhnraEQJKS?usp=sharing The test uses a special case of the holomorphic Wronskian framework introduced in the companion paper. By choosing a specific holomorphic weight function, the complicated boundary oscillations of the zeta function are absorbed into a simpler oscillatory profile. The notebook then plots this profile over a large range of the imaginary part and all phase parameters. If no zero contour appears, the rigidity condition holds numerically, which prevents the simultaneous vanishing of the zeta function off the critical line. Parameters can be modified directly in the notebook.