Dimitrov’s totient–psi–sigma ratio-sum inequality
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Let φ be Euler’s totient, ψ(n)=n∏_{p∣n}(1+1/p), and σ the divisor sum. Put a=φ(n)^k,b=ψ(n)^k,c=σ(n)^k and r=((n−1)/(n+1))^k. Dimitrov’s ratio-sum conjecture asks whether a/(b+c)+b/(a+c)+c/(a+b)≥(4+r+r²)/(2(1+r)) for k≥1,n≥2.
Why it matters and what counts as progress
Why it matters: The ratio-sum inequality refines relations among three multiplicative arithmetic functions beyond the originally proved small-exponent cases.
What counts as progress: Prove the inequality throughout the parameter range and characterize equality, or obtain a stronger composite-integer bound.
How the problem developed
- 10 Jun 2026 · Conjecture [1]
S. I. Dimitrov The paper states conjecture 3, a ratio-sum inequality involving φ, ψ, and σ, as part of a set of related arithmetic inequalities.
OpenAI Codex under Michael Mazur’s direction A Lean repository reports a proof of conjecture 3 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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