Bounded Solutions of a Complex Differential Equation for the Riemann Hypothesis
Walid Oukil
TL;DR
This work recasts the Riemann zeta function zeros as a stability problem for a complex differential equation on $[1,+\infty)$ by introducing $\psi_w(z,t)$ solving $\dfrac{d}{dt} x = w \, t^{-1} x + t^{-1} \{t\}$. It shows that $\zeta(w)=0$ if and only if the corresponding solution with initial value $z=\tfrac{1}{1-w}$ remains bounded, and it identifies a unique bounded solution with $z_w=-\int_1^{\infty} u^{-1-w}\{u\}du$, with a refined asymptotic $\psi_w(z_w,t) = -\dfrac{1}{2w} + O(t^{-1})$. The main theorem equates zeros of $\zeta$ with precise boundedness criteria on $\psi_w(\tfrac{1}{1-w},t)$, translating zero localization into a dynamical systems problem. A conjecture on asymmetry across the critical line is proposed, suggesting that the boundedness of the two related solutions cannot both hold when $\Re(s) \neq \tfrac{1}{2}$, providing a novel analytical lens on the Riemann Hypothesis while acknowledging the indispensable role of the functional equation.
Abstract
In this manuscript, we consider the Riemann zeta function $ζ$, defined through the Abel summation formula. We present a simple analytical method based on a complex differential equation. The aim is to propose a new analytical approach, relying on complex differential equations defined on the interval $[1,+\infty)$, in order to gain insight into the behavior of $ζ(s)$ within the critical strip. We introduce a differential equation depending only on the complex parameter $s$, extracted from the analytical structure of $ζ(s)$ for $s$ in the critical strip. This equation admits a unique continuous and bounded solution. The non-trivial zeros of the zeta function can thus be characterized through the boundedness of such a solution. Furthermore, we conjecture an asymmetry in the boundedness of these solutions with respect to the critical line, suggesting that if $ζ(1-s)= 0$, then $ζ(s) \neq 0$ for any $s$ in the critical strip except on the critical line. This observation does not contradict the Riemann functional equation but supports a formulation consistent with the Riemann Hypothesis, opening a simple yet potentially new direction for the analytical investigation of the zeta function and the localization of its non-trivial zeros.