Bounded Solutions of a Complex Differential Equation for the Riemann Hypothesis

TL;DR

This paper introduces a new complex differential equation approach to analyze the Riemann zeta function within the critical strip, aiming to characterize its zeros through bounded solutions and explore implications related to the Riemann Hypothesis.

Contribution

It proposes a novel analytical method based on a complex differential equation to study the zeta function's zeros, offering a new perspective on the Riemann Hypothesis.

Findings

A differential equation with a unique bounded solution characterizes non-trivial zeros.

Boundedness of solutions may differ across the critical line, indicating potential asymmetry.

The approach suggests a new direction for analyzing the localization of zeros of $\

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.