A family of analogues to the Robin criterion
Steve Fan, Mits Kobayashi, Grant Molnar
TL;DR
This work introduces a κ-parameterized family of divisor-sum analogues σ^{[κ]}(n) built from κ-th LCM-powers, showing σ^{[κ]}(n) ∼ σ(n)^{κ} as κ → ∞ and establishing κ-Robin-type criteria that tie RH to sharp inequalities for all κ > 3/2. The authors develop a theory around κ-colossally abundant numbers via the auxiliary function F^{[κ]}(x, a), obtain mean-value and extremal results for σ^{[κ]}(n), and prove analogues of Robin’s, Ramanujan’s, and Lagarias’s criteria within this κ-framework, including effective κ = 2 results and explicit numerical verifications. The results illuminate how RH implications can be encoded in a family of κ-parametrized inequalities, bridging classical divisor-sum bounds with a broad, structured generalization and suggesting avenues for extensions to other arithmetic functions. Collectively, the paper provides a cohesive framework linking RH to a spectrum of κ-dependent inequalities, with both theoretical and computational backing and clear directions for future refinement and application to related multiplicative functions.
Abstract
The Robin criterion states that the Riemann hypothesis is equivalent to the inequality $σ(n) < e^γn \log \log n$ for all $n>5040$, where $σ(n)$ is the sum of divisors of $n$, and $γ$ is the Euler--Mascheroni constant. Define the family of functions \[ σ^{[k]} (n):=\sum_{[d_1,\dots,d_k]=n}d_1\dots d_k \] where $[d_1, \dots, d_k]$ is the least common multiple of $d_1, \dots, d_k$. These functions behave asymptotically like $σ(n)^k$ as $k\to\infty$. We prove the following analogue of the Robin criterion: for any $k \geq 2$, the Riemann hypothesis holds if and only if $σ^{[k]} (n) < \frac{(e^γn \log \log n)^k}{ζ(k)}$ for all $n > 2162160$, where $ζ$ is the Riemann zeta function.