A family of analogues to the Robin criterion

TL;DR

This paper introduces a family of divisor sum functions generalizing the Robin criterion, establishing an equivalence between the Riemann hypothesis and inequalities involving these functions for all sufficiently large integers.

Contribution

It extends the Robin criterion by defining new divisor sum functions ^{[k]}(n) and proves their inequalities are equivalent to the Riemann hypothesis.

Findings

The functions ^{[k]}(n) asymptotically behave like (n)^k.

The paper establishes a new criterion for the Riemann hypothesis involving ^{[k]}(n).

Inequalities involving ^{[k]}(n) are equivalent to RH for all n > 2162160.

Abstract

The Robin criterion states that the Riemann hypothesis is equivalent to the inequality $\sigma(n) < e^\gamma n \log \log n$ for all $n>5040$, where $\sigma(n)$ is the sum of divisors of $n$, and $\gamma$ is the Euler--Mascheroni constant. Define the family of functions \[ \sigma^{[k]} (n):=\sum_{[d_1,\dots,d_k]=n}d_1\dots d_k \] where $[d_1, \dots, d_k]$ is the least common multiple of $d_1, \dots, d_k$. These functions behave asymptotically like $\sigma(n)^k$ 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 $\sigma^{[k]} (n) < \frac{(e^\gamma n \log \log n)^k}{\zeta(k)}$ for all $n > 2162160$, where $\zeta$ is the Riemann zeta function.