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| Format: | Preprint |
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2020
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| Online Access: | https://arxiv.org/abs/2001.09810 |
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| _version_ | 1866917178732707840 |
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| author | Feng, Nainrong |
| author_facet | Feng, Nainrong |
| contents | Starting from the classical integral representation of the $ζ(s)$ function introduced by Riemann in 1859, this paper reexamines its analytic symmetry structure. By performing a geometric decomposition of the integral representation, we demonstrate that on the critical line $\Re(s)=\frac{1}{2}$, the value of $ξ(s)$ corresponds strictly to the \textbf{real-part projection} of a specific analytic component. This discovery equivalently transforms the problem of complex zeros into a problem of \textbf{sign evolution} along the real axis.
Based on this geometric framework, we \textbf{construct} an analytic mechanism of \textbf{"Two-End Anchoring, Interval Counting"}: the global argument increment on the region boundary \textbf{anchors} the initial value of the phase function, while the geometric decomposition structure on the critical line \textbf{locks} its final value. This mechanism reveals an \textbf{intrinsic coherence} between global topological constraints and local sign oscillations. Unlike traditional methods that rely on asymptotic estimates (such as the Big $O$ error term), the analysis in this paper is \textbf{grounded in} exact identities. It unveils the \textbf{geometric determinism} underlying the zero-counting formula, offering a novel perspective for analytic number theory independent of asymptotic analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2001_09810 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Constraint Structure and Zero Counting in the Integral Representation of the Zeta Function Feng, Nainrong Number Theory 11M06 Starting from the classical integral representation of the $ζ(s)$ function introduced by Riemann in 1859, this paper reexamines its analytic symmetry structure. By performing a geometric decomposition of the integral representation, we demonstrate that on the critical line $\Re(s)=\frac{1}{2}$, the value of $ξ(s)$ corresponds strictly to the \textbf{real-part projection} of a specific analytic component. This discovery equivalently transforms the problem of complex zeros into a problem of \textbf{sign evolution} along the real axis. Based on this geometric framework, we \textbf{construct} an analytic mechanism of \textbf{"Two-End Anchoring, Interval Counting"}: the global argument increment on the region boundary \textbf{anchors} the initial value of the phase function, while the geometric decomposition structure on the critical line \textbf{locks} its final value. This mechanism reveals an \textbf{intrinsic coherence} between global topological constraints and local sign oscillations. Unlike traditional methods that rely on asymptotic estimates (such as the Big $O$ error term), the analysis in this paper is \textbf{grounded in} exact identities. It unveils the \textbf{geometric determinism} underlying the zero-counting formula, offering a novel perspective for analytic number theory independent of asymptotic analysis. |
| title | Constraint Structure and Zero Counting in the Integral Representation of the Zeta Function |
| topic | Number Theory 11M06 |
| url | https://arxiv.org/abs/2001.09810 |