I have a really stupid question that I don't seem to know the answer to and have been too embarassed to ask. In some number theory papers, I encounter sums of the form $$\sum\_{\substack{{x \asymp B}\\P(x) =0}}1$$ to count solutions "of size B" to some quadratic form $P$ (often $x \in \mathbb{Z}^n$ but let us ignore that for now). Alternately, I see statements like let us consider a set of $k$ $\textbf{primes of size $N\log N$}$.


In Vinogradov notation, if $f(x) \ll g(x)$ and $g(x)\ll f(x)$, we write $f(x) \asymp g(x)$, but what does it mean in this context?


For example, when is a prime of size $N\log N$? Vinogradov notation suggests that for a prime
$p \asymp N\log N$, there exist constants $C, C' >0$ such that
$$CN\log N\leq p \leq C'N\log N.$$


In the same vein, $x\asymp B$ should probably mean that there are such positive $k, k'$ such that
$kB\leq |x| \leq k'B$.


My question is: how small and big can these $C$, $C'$, $k$, $k'$ be?


Also, let $\mathcal{P} = \{p \textrm{ prime}: p \asymp N \log N\}$. What does the set $\mathcal{P}$ look like?



What does it mean to have a number of size $B$?
This is a good question, and the answer is that writing $x\asymp B$ under a sum **is sloppy notation without further clarification**. It can mean that $c\_1 B<x<c\_2 B$ for any fixed constants $c\_1$ and $c\_2$ (and then the sum will depend on those constants), or it can mean the same for some constants $c\_1$ and $c\_2$ fixed explicitly at the beginning of the paper. For example, the author might say in the notation section that $x\asymp B$ will mean $B<x<2B$, although some authors would denote this relation by $x\sim B$ to distinguish it from $x\asymp B$.


