# What is the meaning of this notation $O_{\le}(\cdots)$?

I am trying to understand this paper https://faculty.math.illinois.edu/~ford/wwwpapers/primegaps2.pdf

In conclusion of Theorem 3 on page 13, they use the notation $O_{\le}(\delta^{1/10^{J+1}})$. I wonder what does it mean?

• I might add a good proof on prime gaps that's not too difficult is Bertrand's Postulate, which you can find on Wikipedia. – David Reed Dec 4 '17 at 0:19

It has an example: $X = (1+O_{\leq}(\epsilon))Y$ is synomnymous with $(1-\epsilon)Y \leq X \leq (1+\epsilon)Y$
I think he's using big O notation, although I've never seen it written quite that way before. It describes how quickly functions grow in the limit ( basically what they look like really far down the number line). He may just mean $O(\delta...)$. It's consistent in that big Oh notation is frequently used when looking at asymptotic properties of primes.