Equivalence · version 3
RH holds if and only if π(x) = Li(x) + O(√x log x) as x → ∞, where π counts primes and Li denotes the logarithmic integral.
An equivalent formulation of the same canonical question.
Proved
Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.
Bombieri · official RH description · §II, p. 4, equation π(x) = Li(x) + O(√x log x) →
Observed Oct 10, 2026
Replaces an earlier source assessment. Earlier assessment ↓
In this problem · publication v6 · Selected context
Supporting result · Proved
The equivalence is proved. Its two target assertions remain open; proving an equivalence does not prove either target.
The established equivalence joins two open formulations of the same classical question.
The asymptotic estimate has no specified effective constant here.
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.
Open · version 2
π(x) = Li(x) + O(√x log x) as x → ∞.
Scope: An open target equivalent to classical RH, stronger than the prime number theorem.
Relationship statement · Proved · version 3
RH holds if and only if π(x) = Li(x) + O(√x log x) as x → ∞, where π counts primes and Li denotes the logarithmic integral.
Scope: An equivalent formulation of the same canonical question.
Relationship not independently reviewed.
Connection in Riemann hypothesis →Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v13
Supporting result · Proved
The equivalence is proved. Its two target assertions remain open; proving an equivalence does not prove either target.
The established equivalence joins two open formulations of the same classical question.
The asymptotic estimate has no specified effective constant here.
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.
Open · version 2
π(x) = Li(x) + O(√x log x) as x → ∞.
Scope: An open target equivalent to classical RH, stronger than the prime number theorem.
Relationship statement · Proved · version 3
RH holds if and only if π(x) = Li(x) + O(√x log x) as x → ∞, where π counts primes and Li denotes the logarithmic integral.
Scope: An equivalent formulation of the same canonical question.
Relationship not independently reviewed.
Connection in Riemann hypothesis →Relationship not independently reviewed. This concerns its use in this problem.
RH is equivalent to the square-root-scale error term for counting primes.
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 cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.
Observed Oct 10, 2026 · Source passage →
Open · Superseded in an indexed context
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 →