Equivalents to the Riemann Hypothesis

The statement that Robin's criterion holds.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite

@misc{cairn-paper-riemann-hypothesis-equivalents,
  title        = {Equivalents to the Riemann Hypothesis},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/paper-riemann-hypothesis-equivalents}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

robin_criterion_rhs. The statement that Robin's criterion holds.

lagarias_criterion_rhs. The statement that Lagarias' criterion holds.

li_criterion_rhs. The statement that Li's criterion holds.

beurling_nyman_criterion_rhs. The statement that the Beurling-Nyman criterion holds.

weil_positivity_criterion_rhs. The statement that the Weil criterion holds.

This file contains formal statements that are known to be equivalent to the Riemann Hypothesis, following the survey [Co26], §4. The statement of the Riemann Hypothesis itself is in FormalConjectures.Millennium.RiemannHypothesis.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Paper.RiemannHypothesisEquivalents (5 statements).

theorem robin_criterion_rhs {n : ℕ} (hn : n > 5040) :
    σ 1 n < exp eulerMascheroniConstant * n * log (log n)

theorem lagarias_criterion_rhs {n : ℕ} (hn : 1 < n) :
    σ 1 n < harmonic n + exp (harmonic n) * log (harmonic n)

theorem li_criterion_rhs {n : ℕ} (hn : 0 < n) : 0 < lambda n

theorem beurling_nyman_criterion_rhs :
    Dense { fractScalarInvSumL2 n c θ | (n) (c) (θ)
      (hθ : ∀ ν, θ ν ∈ Set.Ioc 0 1) (h : ∑ ν, c ν * θ ν = 0) }

theorem weil_positivity_criterion_rhs {g : C_c(ℝ, ℝ)} (hg : ContDiff ℝ ∞ g)
    (h : IsTestFunction g) :
    weilArchimedean (g ⋆ fun x ↦ g (-x)) +
      ∑' p : Nat.Primes, weilNonArchimedean p (g ⋆ fun x ↦ g (-x)) ≤ 0

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

[Co26] A. Connes, [The Riemann Hypothesis: Past, Present and a Letter Through Time](https://arxiv.org/abs/2602.04022), 2026.

Source and licence

Imported from Formal Conjectures (research papers), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.