Bounded Solutions of a Complex Differential Equation for the Riemann Hypothesis
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866912688334962688 |
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| author | Oukil, Walid |
| author_facet | Oukil, Walid |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_05521 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Bounded Solutions of a Complex Differential Equation for the Riemann Hypothesis Oukil, Walid General Mathematics 00A05, 34A30, 34E05 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. |
| title | Bounded Solutions of a Complex Differential Equation for the Riemann Hypothesis |
| topic | General Mathematics 00A05, 34A30, 34E05 |
| url | https://arxiv.org/abs/2112.05521 |