# What does this $\asymp$ symbol mean? (subject: analytic number theory)

I'm reading a survey article by Andrew Granville on analytic number theory.

On page 22 of the paper, there appears a strange looking symbol, undefined. I've circled it in red in the screenshot below. Since it's not defined in the paper, I'm assuming it must be standard notation.

From the context, I'm assuming it means something like "as compared to", or "with reference to", but that's just a guess.

Can anyone identify the symbol, even better explain what it means and/or provide a reference? Is there a name to speak the symbol?

It can mean different things depending on the context. For instance, in Graham, Knuth, and Patashnik's Concrete Mathematics it's defined to mean the same thing as "Big $\Theta$" (see p. 448), as in

$$f \asymp g \iff \exists\, C,D>0 : C|g| \leq |f| \leq D|g|,$$

but I read a paper recently where it was instead defined to mean the same thing as $\sim$ (as defined here).

• Yes, I think you're right. Having looked at your LaTeX code, I see the symbol is \asymp, and doing a search brings up this useful table on asymptotic notations that confirms your (nicer) expression above. (I'll paste the table into a separate answer below for reference.) Accepting your answer as the correct one. Thanks! – Assad Ebrahim Apr 22 '14 at 20:44
• (+1)... in English: "there exist positive constants C,D such that $f$ can be sandwiched between $g$ scaled appropriately above and below." – Assad Ebrahim Apr 22 '14 at 20:56

Searching further on Antonio Vargas's accepted answer above finds an insightful short paper of A.J. Hildebrand on Asymptotic Notation.

From this the following useful reference table is screenshotted below: 