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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?

tyobrien
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  • $a<b$ doesn't make sense for either $a$ or $b$ being a nonreal complex number. – Gerry Myerson Oct 29 '18 at 02:19
  • Not so. See the links. – tyobrien Oct 29 '18 at 02:22
  • I've been doing mathematics for 50 years. I don't have to see any links. I know that neither $7\le8+9i$ nor $7\ge8+9i$ makes any sense. – Gerry Myerson Oct 29 '18 at 02:29
  • If you check out the definition of an ordered field (say, at https://en.wikipedia.org/wiki/Ordered_field), you'll find that you cannot "impose an ordering of the complex numbers", not unless you are willing to give up one or more of the properties that make an order an order. You can ask for a function $f$ with $\sigma(n)<|f(n)|<e^{\gamma}n\log\log n$, but then you're really asking about the real-valued function $g(n)=|f(n)|$, and you haven't gained anything. – Gerry Myerson Oct 30 '18 at 04:46

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