The Riemann hypothesis as a statement about natural numbers

Question

In this answer it is claimed that the Riemann hypothesis can be expressed as a statement about natural numbers. How would that look like?

Answer 1

He's probably referring to either Robin's theorem (most likely), which establishes that Riemann hypothesis is equivalent to the fact that for all $n\ge5041$, $$\sigma(n)<e^\gamma n\log\log n$$ where $\sigma(n)=\sum_{d\mid n} d$ and $\gamma$ is Euler-Mascheroni constant, or to this result by Lagarias which establishes that Riemann hypothesis is equivalent to the statement that, for all $n\ge2$, $$\sigma(n)<H_n+e^{H_n}\log H_n$$

where $H_n=\sum_{k=1}^n\frac1k$.

Added: For instance, refer to the Theorem 3.2 in Lagarias' paper (an improvement on prop. 1 section 4 of Robin's paper):

The negation of Riemann hypothesis implies that there are some $C>0$ and some $0<\beta<\frac12$ such that $\frac{\sigma(n)}{n\log\log n}\ge e^\gamma+\frac C{(\log n)^\beta}$ holds frequently as $n\to\infty$.

As a consequence, Riemann hypothesis is false if and only if there is some integer $m\ge5041$ (in fact, infinitely many) such that $\sigma(m)>e^\gamma m\log\log m$. A purported such number $m$ may be (theoretically) checked to satisfy this inequality in finitely many passages of arithmetic involving rational numbers.