feat: formal statements of Robin's and Lagarias' inequalities equivalent to RH

[cascone26]

Summary

This PR adds Mathlib/NumberTheory/LSeries/RobinInequality.lean, which contains formal Lean 4 statements of two classical equivalences of the Riemann Hypothesis:

Robin's theorem (1984): RH ↔ σ₁(n) < e^γ · n · ln(ln(n)) for all n > 5040

Lagarias' theorem (2002): RH ↔ σ₁(n) ≤ H_n + exp(H_n) · ln(H_n) for all n ≥ 1

New declarations

• robinBound (n : ℕ) : ℝ — the Robin bound e^γ · n · ln(ln(n))
• lagariasBound (n : ℕ) : ℝ — the Lagarias bound H_n + exp(H_n) · ln(H_n)
• robin_iff_RH : RiemannHypothesis ↔ ∀ n > 5040, σ₁(n) < robinBound n
• lagarias_iff_RH : RiemannHypothesis ↔ ∀ n ≥ 1, σ₁(n) ≤ lagariasBound n

Status

The statements are complete and use existing Mathlib primitives:

• ArithmeticFunction.sigma for σ₁
• Real.eulerMascheroniConstant for γ
• harmonic for H_n (with ℚ → ℝ coercion)
• RiemannHypothesis from Mathlib.NumberTheory.LSeries.RiemannZeta

The proofs are currently sorry. The Robin direction (RH → inequality) requires analytic number theory following Robin's original 1984 paper. The converse uses Gronwall's theorem on superior highly composite numbers. Both are substantial Mathlib projects in their own right.

This PR establishes the formal statements as a foundation for future proof work. These equivalences are not currently formalized anywhere in Mathlib.

References

• G. Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. 63 (1984), 187–213.
• J. Lagarias, An elementary problem equivalent to the Riemann hypothesis, Amer. Math. Monthly 109 (2002), 534–543.

Note on sorry

I'm aware Mathlib does not merge sorry-laden PRs in general. I'm submitting this as a draft to:

1. Get feedback on whether the statement formulation is idiomatic
2. Establish that these statements don't already exist in Mathlib under a different name
3. Invite collaboration on completing the proofs

[github-actions]

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[github-actions]

PR summary 6ef8cc2731

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.NumberTheory.LSeries.RobinInequality (new file)	2719

---

Declarations diff

+ RH_of_lagarias_inequality
+ RH_of_robin_inequality
+ lagariasBound
+ lagarias_iff_RH
+ lagarias_inequality_of_RH
+ robinBound
+ robin_iff_RH
+ robin_inequality_of_RH

You can run this locally as follows

## summary with just the declaration names:
./scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh contains some details about this script.

---

No changes to technical debt.

You can run this locally as

./scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).