How to study analytic number theory? Should i study many books at the same time? Or should i go one by one? I started with Apostol's book... But for example if i don't understand a proof, i check another book for a different proof of the same theorem and so on... It's a little distracting to be fair... Is it wrong to study this way?
I'm looking for advices for studying math generally. I'm always taking notes for what i read. Again and again. I'm trying to memorize all the proofs. Is this wrong? It's not like i'm memorizing it without understanding.
Unnecessary question: Should i own a whiteboard for studying math? Is it helpful? :)
 A: First of all, it will help if you  tell us what mathematical background you have.
For example, if you don't know the basics from elementary number theory then it is not such a good idea to try analytic number theory.
My advice is the following:    
If it is not possible to study mathematics and you are self-taught:    
1.Post your questions here in mathSE and try to answer other questions on your own(not neccesarily posting them) because it is important to be a part of a mathematical community
Nobody has ever done anything on his own (even Ramanujan communicated with Hardy)  
2.Analytic Number Theory is a difficult part of Number Theory.I would advise you to start reading something more "human" first, such as Calculus or Elementary Number Theory.
Closing, I do not have the best opinion for Apostol's book, so don't get stuck there.
Try other books also.(But this just my opinion)
A: PART ONE: I am not sure of what background you have, but I think you need some amount of basic complex analysis to comfortably handle Apostol's book. If I were to give you a proper structure, I would tell you to pick up Complex Variables first. You could use a non-rigorous text like Brown and Churchill's Complex Variables and their Applications, or a more serious text, like Conway's Functions of One Complex Variable: Part I, or Lars Ahlfors's Complex Analysis. Gamelin's Complex Analysis is also a nice read, I've heard, but haven't read it myself. 
Then, Apostol's 'Introduction to Modern Analytic Number Theory' is a nice book to start off with. You could also simultaneously look at 'Problems in Analytic Number Theory' by Ram Murty. After you are done with that, Davenport's 'Multiplicative Number Theory' is a standard book to follow. I liked a little book by Mendes France and Tenenbaum, titled 'The Prime Numbers and Their Distribution'. 
Other recommendations I have heard of are J.P. Serre's 'A Course in Arithmetic', Jameson's 'The Prime Number Theorem', and a follow-up to his first book by Apostol on modular forms.
Beyond this, I think, if you know some basic Fourier Series, you can go ahead and read 'Analytic Number Theory' by Iwaniec and Kowalski, an encyclopedic account of Analytic Number Theory.  
Once you have finished most of this, Iwaniec and Kowalski recommend that you pick up some knowledge of Automorphic Forms, Algebraic Number Theory, and Algebraic Geometry in the preface of their text. 
PART TWO: You should try proving the results on your own, but if you are really stuck after several sincere attempts, reading a proof isn't wrong. Remembering a proof is often over-rated. I think that remembering a nice strategy of a particular proof could help with other proofs later, but knowing all the proofs at your fingertips is unnecessary. I think it is better to just prove things on your own if you can, and if you aren't able to, read the proof, and if the technique is something worth remembering, then, do so. Taking notes is a nice strategy of maintaining a log of what you've learnt, but can be very time-consuming. But, really, it depends on what you think works for you. 
PART THREE: I personally prefer chalkboards, but a big whiteboard often comes in handy.  
