Analysis of the Riemann Zeta Function via Recursive Taylor Expansions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908874885300224 |
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| author | Bai, Yunwei |
| author_facet | Bai, Yunwei |
| contents | We present an unconditional proof that non-trivial zeros of the Riemann Zeta function must lie strictly on the critical line $\text{Re}(s) = 0.5$. By defining a recursive path of Taylor expansions originating from the domain of absolute convergence, we translate the zeta function towards the critical region, which is an easy-to-understand form of the analytical continuation. We then assume the existence of off-critical-line (off-line) zeros, which exist in pairs symmetric by the critical line. If the pairs are zero in value, their real and imaginary components differences should be both zero. However, we derive a contradiction against the assumption via basic logical deduction, proving the non-existence of the off-line zeros. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_05122 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Analysis of the Riemann Zeta Function via Recursive Taylor Expansions Bai, Yunwei General Mathematics We present an unconditional proof that non-trivial zeros of the Riemann Zeta function must lie strictly on the critical line $\text{Re}(s) = 0.5$. By defining a recursive path of Taylor expansions originating from the domain of absolute convergence, we translate the zeta function towards the critical region, which is an easy-to-understand form of the analytical continuation. We then assume the existence of off-critical-line (off-line) zeros, which exist in pairs symmetric by the critical line. If the pairs are zero in value, their real and imaginary components differences should be both zero. However, we derive a contradiction against the assumption via basic logical deduction, proving the non-existence of the off-line zeros. |
| title | Analysis of the Riemann Zeta Function via Recursive Taylor Expansions |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2603.05122 |