This paper introduces a new, finite form of Robin’s inequality, a statement that links the Riemann hypothesis to how large the sum of an integer’s divisors can be. If σ(n) denotes the sum of all positive divisors of n, the authors propose the inequality

σ(n)/n < e^γ · Σ_{j=0}^{ω(n)} ((ln ln ln n)^j / j!), for n > 5040,

where ω(n) is the number of distinct prime factors of n and γ is the Euler–Mascheroni constant. In plain terms, the bound uses information about how many different primes divide n and a short (finite) truncation of an exponential series built from iterated logarithms of n.

What the authors did. They derived this arithmetic, truncated version of Robin’s inequality and showed it is pointwise stronger than the classical Robin bound. Concretely, they prove the new inequality unconditionally for several important classes of integers: those with at most six distinct prime factors (ω(n) ≤ 6), primorials (numbers made by multiplying together the first few primes), odd integers, and square-free integers (numbers with no repeated prime factors).

Why this matters. Robin’s inequality is known to be equivalent to the Riemann hypothesis (RH), a central unsolved problem in number theory. The authors prove that their truncated inequality is equivalent to Robin’s inequality and therefore also equivalent to RH. One direct consequence they highlight is that RH is equivalent to the truncated inequality holding on all colossally abundant numbers — a special class of integers that tend to maximize divisor sums.

Important caveats and limits. The new inequality is proven only for the specific classes listed above; the general case for all integers is not resolved in this paper. Because the truncated form is equivalent to Robin’s inequality, proving it for all n would be as hard as proving the Riemann hypothesis itself. The authors also show that if the truncated inequality ever fails, the smallest counterexample must be a superabundant number — another rigid, extremal class of integers that frequently appear in divisor-sum problems. The statements apply only for n > 5040.