Theorem · version 2
For a smooth projective geometrically connected curve C/F_q of genus g, Z_C(T) = P_C(T)/((1−T)(1−qT)), with P_C(T)=∏_{j=1}^{2g}(1−α_j T) and |α_j|=√q under every complex embedding.
Weil’s RH analogue for curves over finite fields.
Proved
Only the exact statement, domain and revision cited by this Claim; this is not a resolution of classical RH or independent OpenMath expert assessment.
Bombieri · official RH description · §IV, pp. 6–8: curve zeta functions and Weil’s theorem →
Observed Oct 10, 2026
In this problem · publication v7 · Selected context
Different statement · related variant · Proved
This is proved in another mathematical domain. Its geometric proof gives insight but does not establish classical RH; it is not counted as a proved special case of ζ.
The finite-field curve theorem places Frobenius eigenvalues on the expected circle, paralleling a zero-location assertion.
Its function field, cohomology and variable differ; it is not proved classical RH.
Open · version 2
Every nontrivial zero ρ of the meromorphic continuation of ζ(s) has Re(ρ) = 1/2.
Scope: Every zero at every height; negative-even trivial zeros and the pole at 1 are excluded.
Proved · version 2
For a smooth projective geometrically connected curve C/F_q of genus g, Z_C(T) = P_C(T)/((1−T)(1−qT)), with P_C(T)=∏_{j=1}^{2g}(1−α_j T) and |α_j|=√q under every complex embedding.
Scope: Weil’s RH analogue for curves over finite fields.
Relationship not independently reviewed.
Connection in Riemann hypothesis →Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v13
Different statement · related variant · Proved
This is proved in another mathematical domain. Its geometric proof gives insight but does not establish classical RH; it is not counted as a proved special case of ζ.
The finite-field curve theorem places Frobenius eigenvalues on the expected circle, paralleling a zero-location assertion.
Its function field, cohomology and variable differ; it is not proved classical RH.
Open · version 3
Every nontrivial zero of the meromorphic continuation of has .
Scope: Every zero at every height; negative-even trivial zeros and the pole at 1 are excluded.
Proved · version 2
For a smooth projective geometrically connected curve C/F_q of genus g, Z_C(T) = P_C(T)/((1−T)(1−qT)), with P_C(T)=∏_{j=1}^{2g}(1−α_j T) and |α_j|=√q under every complex embedding.
Scope: Weil’s RH analogue for curves over finite fields.
Relationship not independently reviewed.
Connection in Riemann hypothesis →Relationship not independently reviewed. This concerns its use in this problem.
The finite-field curve analogue has been proved.
paraphrase · No upstream full text or code is republished by this record.
Do all nontrivial zeros of the Riemann zeta function have real part 1/2?
paraphrase · No upstream full text or code is republished by this record.
Enrico Bombieri · Published
Official Clay exposition. Historical results and approaches must not be read as current record bounds.
Source revision: official-description. Observed Oct 10, 2026.
The cited work does not report AI involvement; absence of attribution is not evidence of absence.
Official description PDF: official-description. External source evidence; no OpenMath execution or independent reproduction is attached.
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Proved
Only the exact statement, domain and revision cited by this Claim; this is not a resolution of classical RH or independent OpenMath expert assessment.
Observed Oct 10, 2026 · Source passage →