Robin Extrema and Counterexample Support Bounds

Current state

The main manuscript characterizes the integer and real-exponent optimizers when the number of distinct prime factors is fixed and at least six. It evaluates the gap between their logarithmic objective values asymptotically and gives a finite search for the integer maximum, assuming exact comparisons. The search uses the classical exponent ordering for colossally abundant numbers. Finite computations settle the small-support cases and the transition at $m=30$ in the problem $\omega (n)\le m$. The interval implementation certifies the reported cases and stops if it cannot resolve a comparison.

A separate manuscript proves unconditionally that any counterexample to Robin’s inequality has more than $3.34\times 10^{33}$ distinct prime factors. The proof uses elementary size bounds, Polak’s primewise decomposition, and explicit estimates for primes and zeros of the zeta function. A certificate computed with interval arithmetic verifies the numerical inequalities.

Future plans

I want to turn the finite characterization into a general numerical solver with proved comparison and termination guarantees, make the rounding estimates effective, and complete the comparison with earlier work on superabundant and colossally abundant numbers. I am also interested in the corresponding extremal problems for other multiplicative functions and for Beurling prime systems.