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Bounded Solutions of a Complex Differential Equation for the Riemann Hypothesis

21 July 2025, Version 23
Working Paper
This content is an early or alternative research output and has not been peer-reviewed by Cambridge University Press at the time of posting.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

In this paper, we consider the representation of the Riemann zeta function $\zeta$ defined by Abel's summation formula. Using the differential equations, we show that: if $|\zeta(s)|=0$ then $|\zeta(1-s)|\neq0$ for any point $s$ in the critical strip except the critical line.

Keywords

Zeta function
Non trivial zeros
Riemann hypothesis

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Version History

Jul 21, 2025 Version 23

Version Notes

An analytical proof has been employed in this version. The numerical results indicate that the zeta function is not symmetric, as stated in this manuscript.

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License

All Rights Reserved

DOI

10.33774/coe-2023-sb63n-v23
D O I: 10.33774/coe-2023-sb63n-v23 [opens in a new tab]

Author’s competing interest statement

The author(s) have declared they have no conflict of interest with regard to this content

Ethics

The author(s) declare that they have sought and gained approval from the relevant ethics committee/IRB for this research and its publication.

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