Statement copy in the gpt 6 astra pro collection (Math-2)

The Riemann Hypothesis

Definitions and mathematical statement

For $s\in{\mathbb{C}}$ with $\Re s>1$, define \[ \zeta(s)=\sum_{n=1}^{\infty}n^{-s}=\prod_{p\text{ prime}}(1-p^{-s})^{-1}. \] Use its meromorphic continuation to ${\mathbb{C}}$, whose only pole is at $s=1$. The assertion is \[ \forall\rho\in{\mathbb{C}},\qquad \left(0<\Re\rho<1\ \text{and}\ \zeta(\rho)=0\right) \ \Longrightarrow\ \Re\rho=\tfrac12. \] The open critical strip excludes the negative even integer zeros. The assertion concerns locations, not multiplicities, of zeros.

Formulation and concept references: [S1].

Short English statement

Are all zeta zeros between the vertical lines zero and one located on the halfway line?

Sources

• S1 Enrico Bombieri, Problems of the Millennium: the Riemann Hypothesis, Section I and the prime-counting equivalence. https://www.claymath.org/wp-content/uploads/2022/05/riemann.pdf. Catalogue formulation source. Consulted 24 September 2026: Section I, definition and zero-location statement.
• S2 Riemann Hypothesis. https://www.claymath.org/millennium/riemann-hypothesis/. Catalogue research/status reference; not a proof endorsement.
• S3 A new proof that more than 2/3 of the zeros of the Riemann zeta function are simple and on the critical line. https://arxiv.org/abs/2609.02882. Catalogue research/status reference; not a proof endorsement.
• S4 Lamzouri lecture at Max Planck Institute for Mathematics, September 3, 2026. https://math-events.uni-bonn.de/event/1560/. Catalogue research/status reference; not a proof endorsement.
• S5 On the Extended, Generalized, and Grand Riemann Hypotheses: A Unified Approach Based on the General Properties of L-Functions — Weicun Zhang, v19. https://www.preprints.org/manuscript/202506.0481. Catalogue research/status reference; not a proof endorsement.

• E1 Dave Platt and Tim Trudgian, The Riemann hypothesis is true up to $3\cdot10^{12}$, arXiv:2004.09765v1 (21 April 2020), Theorem 1 and Section 2; published in Bulletin of the London Mathematical Society $2021$. https://arxiv.org/pdf/2004.09765v1. Research reference: rigorous bounded-height verification, including completeness of the zero count. Consulted 24 September 2026.

• E2 Larry Guth and James Maynard, New large value estimates for Dirichlet polynomials, Annals of Mathematics 203 $2026$, no. 2, 623--675, DOI: 10.4007/annals.2026.203.2.6, Theorem 1.2. https://annals.math.princeton.edu/2026/203-2/p06. Author manuscript: arXiv:2405.20552v2 (7 April 2026), https://arxiv.org/pdf/2405.20552v2. Research reference: zero-density estimate; publication metadata and the stated theorem were checked. Consulted 24 September 2026.

• E3 Michael Griffin, Ken Ono, Larry Rolen and Don Zagier, Jensen polynomials for the Riemann zeta function and other sequences, arXiv:1902.07321v2 (31 March 2019), Section 1, Theorems 1--2; published in Proceedings of the National Academy of Sciences $2019$. https://arxiv.org/pdf/1902.07321v2. Research reference: the fixed-degree, sufficiently-large-shift quantifiers, not all-degree hyperbolicity. Consulted 24 September 2026.

• E4 Brad Rodgers and Terence Tao, The de Bruijn--Newman constant is non-negative, Forum of Mathematics, Pi 8 $2020$, e6, DOI: 10.1017/fmp.2020.6. Author manuscript arXiv:1801.05914v5 (3 July 2021), Section 1, equations $1$--$4$, and the main theorem. https://arxiv.org/pdf/1801.05914v5. Research reference: the theta-kernel normalization and the endpoint obstruction. Consulted 24 September 2026.

• E5 David W. Farmer, Jensen polynomials are not a plausible route to proving the Riemann Hypothesis, arXiv:2008.07206, Sections 2--4. https://arxiv.org/pdf/2008.07206. Research reference: differentiation and information loss; the paper's methodological assessment is not used as an impossibility theorem. Consulted 24 September 2026.

• E6 Jonathan Holland, A new hyperbolicity wedge and a joint semicircle limit for Jensen polynomials of Riemann's $\xi$-function, arXiv:2608.08682v1 (9 August 2026), Theorem 1.1. https://arxiv.org/html/2608.08682v1. Research reference: recent manuscript, statement inspected, full proof not independently audited here; not used as a premise of any lemma. Consulted 24 September 2026.

• E7 Kiran S. Kedlaya, More on the zeroes of $\zeta$, MIT 18.785: Analytic Number Theory, spring 2007, Section 1, Lemma 1 and Theorems 2--3. https://kskedlaya.org/18.785/zeroes.pdf. Research reference: growth of the completed function, zero counting, and Hadamard factorization. Consulted 24 September 2026.

• E8 Paul Garrett, 12. Weierstrass and Hadamard products, author lecture notes, 3 February 2021, Section 3 (Hadamard products). https://www-users.cse.umn.edu/~garrett/m/complex/notes_2020-21/12_Hadamard_products.pdf. Research reference: canonical products and the finite-order factorization theorem. Consulted 24 September 2026.

• E9 Youness Lamzouri, A new proof that more than $2/3$ of the zeros of the Riemann zeta function are simple and on the critical line, arXiv:2609.02882v2 (8 September 2026), Theorem 1.1, Proposition 2.1 and Sections 2--3. https://arxiv.org/html/2609.02882v2. Version-specific research supplement to [S3]: the stated bound and method were inspected. Neither a complete analytic proof audit nor a formal-certificate audit was performed. Consulted 24 September 2026.