# First use of ~ and ≍ (\sym and \asymp)

The relations ~ and ≍ are frequently used in math and computer science, at least within number theory and analysis of algorithms. What is their origin?

# Definitions

Suppose $$g(x)$$ is an eventually-positive function of $$x$$. Then

$$f(x) \sim g(x)$$

if and only if

$$\lim_{x\to\infty} f(x)/g(x) = k$$ with $$0 while

$$f(x) \asymp g(x)$$ if and only if $$0 < \liminf_{x\to\infty} f(x)/g(x) \le \limsup_{x\to\infty} f(x)/g(x) < \infty.$$

Note that the defintion of ~ replaces the $$\le$$ for $$=$$ in the above.

# Related work

I believe the related notations $$\ll$$ and $$\gg$$ are due to Vinogradov, while $$O/o/\Omega/\omega$$ are due to Hardy and Littlewood. (I've skimmed through Hardy's Orders of Infinity which suggests that he and Paul Du Bois-Reymond did some early work here but I don't think that either of these notations originated there.)

I was not able to find these in either of these online resources:

The notation $$\sim$$ does not mean that ratio $$f(x)/g(x)$$ has some positive limit $$k$$, but that it has limit $$1$$, e.g., $$2x + \sqrt{x} \sim 2x$$. What math books do you know that define $$\sim$$ in the way you did? It would allow us to write $$2x+\sqrt{x} \sim x$$, which is a meaning of $$\sim$$ that I've never seen.
This notation $$\sim$$ is due to Landau: see page 62 of the first volume of his Handbuch der Lehre von der Verteilung der Primzahlen (1909). It's available at archive.org here.
The $$O$$-notation and $$o$$-notation are not due to Hardy and Littlewood:
1. The $$o$$-notation is due to Landau in the same book I already mentioned, on page 61.
2. The $$O$$-notation is due to Bachmann on p. 401 of his book Die Analytische Zahlentheorie (1894) here.
• Probably $\Omega$ and $\omega$ are more recent than Hardy and Littlewood. Jul 16, 2023 at 1:13