How can I get more comfortable working with integrals and bounds in analytic number theory?

I've started self-studying analytic number theory from these lecture notes (I am currently attempting exercises from Chapter 2), and even though I enjoy the learning process, my main difficulty seems to be my intuition regarding bounds and asymptotic analysis. I got through introductory real analysis pretty well because I sorta knew what I was trying to bound, but here I am completely lost, to the point where I have to read each proof line-by-line again and again to develop an intuition about what the proof is trying to do. I know that the usual answer to these kinds of questions would be just to do more problems, but when I look at the problems (especially the later ones), I don't have a slightest idea where to start. This is almost demotivating because I am sure ANT has a lot of beautiful ideas and proofs rather than just bounding stuff endlessly, but it seems like the journey to get to those ideas is going to be painstakingly long. Therefore my questions would be:

1. How can I get more comfortable working with bounds and integrals? Is it a thing that gradually comes with experience?
2. Should I set aside my urge for developing intuition until a later time?

Thanks a lot!

• i think i do get what you mean, but still to be sure can you give one or two examples of proofs in that book where you found integrals and bounds overwhelming and lost sight of the big picture? Apr 2, 2022 at 12:39
• @tomos Hi! I think the first time I got overwhelmed is in the proof of theorem 2.13 where the author shows that if ordinary mean value exists, then logarithmic mean value also exists and is equal to the ordinary one, around the part where they try to show that $\lim_{x \to \infty} \frac{I(x)}{\log x} = A$. Apr 2, 2022 at 12:58
• Learn summation by parts very well as it is the crux of many estimates Apr 2, 2022 at 14:38
• I would say practicing is very helpful as ANT is an art of bounding Apr 2, 2022 at 16:31
• hmm ok. maybe writing $\log x$ seemed a bit random? i guess there's not much to say - seeing $\log$s and $1/t$'s interchange is a thing you see often, and you just sort of get used to it. sorry i know that's not very helpful:/ the rest in that proof is "just" definitions though, right? but i do know that just one thing in a proof is enough to make it all seem foreign. it's helpful often to write an "outline" of the proof. (like, write three or four steps describing the proof just in everyday words, no maths, just things like "first find a bound for A" then "find one for B" then "compare") Apr 3, 2022 at 22:04