Research paper

The Integrality Gap in Robin's Inequality

A study of the extremal value of Grönwall’s ratio $σ (n) / (n lo g lo g n)$ among integers with a fixed number of distinct prime factors, and of the relaxation obtained by allowing real exponents.

In progress Last substantially updated August 15, 2026

Current state

Consider the maximum of $σ (n) / (n lo g lo g n)$ as a function of the exponents in the prime factorization of $n \geq 5041$ , with finite support constraints. This invites a domain relaxation by letting the exponents be real-valued rather than integer-valued. Currently, we characterize the maximum for both the integer and relaxed problems and evaluate the integrality gap asymptotically. We relate the fixed-support formulation back to Robin’s inequality, settle the small-support cases with a finite-computation supplement, and characterize the budget-constrained problem $ω (n) \leq m$ , including its transition at $m = 30$ .

Future plans

I would like to inspect more carefully the relation to SA/CA numbers and investigate what happens under perturbations to the problem statement, e.g., studying the extremal growth of some other multiplicative functions, or using Beurling prime systems instead of the primes.

Details

Phase: Public manuscript
Domain: Analytic number theory · discrete optimization · computational verification
My role: Author

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