I have a problem where I need to compare real and complex numbers. I see here and here that there are different ways to go about interpreting the sizes of complex numbers, but in my context I want to see if this is allowed.
To be precise, Robin's Inequality which is equivalent to the Riemann Hypothesis is
$$\sigma(n)\lt e^\gamma n\log\log n$$ for all $n\gt 5040$.
Suppose we want to include a comparison with the function $f:\mathbb{N}\to\mathbb{C}$ that looks similar to
$$\sigma(n)\lt f(n) \lt e^\gamma n\log\log n.$$
So can we impose an ordering of the complex numbers that preserves Robins Inequality’s equivalence to the Riemann Hypothesis and allows for the statement above under such an ordering?