Dimitrov’s totient–psi–sigma product inequality
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Let φ(n) count integers coprime to n, ψ(n)=n∏_{p∣n}(1+1/p), and σ(n) be the divisor sum. For integers k≥1,n≥2, Dimitrov’s product conjecture asks whether (φ(n)^k+ψ(n)^k+σ(n)^k)(φ(n)^(−k)+ψ(n)^(−k)+σ(n)^(−k))≥5+2((n+1)/(n−1))^k+2((n−1)/(n+1))^k.
Why it matters and what counts as progress
Why it matters: The full parameter range extends the known k=1 arithmetic-function inequality and identifies whether prime integers are the only equality cases.
What counts as progress: Prove the full k,n range and the prime-only equality cases, or sharpen the bound for composite n.
How the problem developed
- 10 Jun 2026 · Conjecture [1]
S. I. Dimitrov The paper states conjecture 1, a lower bound for an expression involving φ, ψ, and σ, with its original discussion focused on the k=1 case.
OpenAI Codex under Michael Mazur’s direction A Lean repository reports a proof for every k≥1 and n≥2, with equality exactly when n is prime; the project was registered with the Palomar Formal Proof Registry.
AI contribution
Codex proves two arithmetic-function conjectures in Lean
Lean proofs establish Dimitrov’s Conjectures 1 and 3, the stated rational lower bounds involving Euler’s totient φ(n), Dedekind’s ψ(n) and σ(n), for every integer k≥1 and n≥2. Equality in each bound holds exactly when n is prime.
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