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This is a Quantitative Study study.

September 28, 2025Open Access

Bounded Solutions of a Complex Differential Equation for the Riemann Hypothesis

Key Points

The study demonstrates |(1-s)ζ(s) -ȳζ(1-ȳ)| ≠ 0 in the critical strip, confirming non-trivial zeros lie on the critical line.
Results align with the Riemann Hypothesis, reinforcing the structure of the Riemann zeta function in relation to the critical line.
The findings highlight an asymmetry across the critical strip, supporting claims about the non-existence of off-line zeros.
This analysis uses differential equations to delve into the nature of the Riemann zeta function under Abel's summation formula.

Abstract

In this paper, we consider the representation of the Riemann zeta function defined by Abel's summation formula. Using the differential equations, we show that | (1-s) (s) -s (1-s) | 0 for any point s in the critical strip except the critical line. This result suggests an asymmetry of (1-s) (s) across the critical strip. This does not contradict the Riemann functional equation but prove that non-trivial zeros cannot lie off the critical line. These results are consistent with the Riemann Hypothesis and suggest that non-trivial zeros lie on the critical line.

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