Bounded Solutions of a Complex Differential Equation for the Riemann Hypothesis

Abstract

In this manuscript, we consider the Riemann zeta function $\zeta$, 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 $\zeta(s)$ within the critical strip. We introduce a differential equation depending only on the complex parameter $s$, extracted from the analytical structure of $\zeta(s)$ 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 $\zeta(1-s)= 0$, then $\zeta(s) \neq 0$ 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.