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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?

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    $\begingroup$ I might add a good proof on prime gaps that's not too difficult is Bertrand's Postulate, which you can find on Wikipedia. $\endgroup$ – David Reed Dec 4 '17 at 0:19
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The notation is described on page 7 of the pdf.

It has an example: $X = (1+O_{\leq}(\epsilon))Y$ is synomnymous with $(1-\epsilon)Y \leq X \leq (1+\epsilon)Y$

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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.

Check out this article and see if it fits:

https://en.wikipedia.org/wiki/Big_O_notation

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