Could anyone tell me in what sense the following is an "asymptotic formula":


Theorem 1 from


[Link](https://projecteuclid.org/journals/michigan-mathematical-journal/volume-17/issue-1/Primes-in-arithmetic-progressions/10.1307/mmj/1029000373.full)


(this is open access, so I think I'm allowed to link it)


At the moment, I'm just trying to understand what the bit with $Q\leq x$ means. I assume the Theorem says something like: the LHS varies like the main term, in that LHS/[main term] tends to one as x tends to infinity. But it's not clear to me that [error terms]/[main terms] tend to zero. There seems to be definitely something I'm not understanding with the relationship between $Q$ and $x$.


Can anyone clarify?


Thanks very much.



Asymptotic formula in Analytic Number Theory
There is no Theorem 1 in this paper. The statement called "Theorem" contains an asymptotic formula: e.g. (2) has a main term $Qx\log x$ and two error terms $O(\dots)$. The error terms are $o(Qx\log x)$ for $x (\log x)^{-A} < Q < x $ and $x\to\infty$, say, hence in this case the left hand side is asymptotic to the main term.


