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I am a person living in a 3rd world country and completed my masters in mathematics in July 2020. Then I began to study some additional topics in Pure Mathematics as I was applying for Ph.D. abroad( In Number Theory) as I felt that I should study more topics before applying in December2020- January 2021.

Background in analytic number theory: I have studied Number Theory from Elementary number theory by Burton, Introduction to Analytic Number Theory (Tom Apostol) , Modular Functions and Dirichlet Series in Number Theory ( Tom Apostol).

I am thinking of studying more Analytic number theory in the free time I am having. But instead of studying from textbook I thought of studying from Lecture notes ( If available on course page of course) as when studying from textbook one covers many more topics than required in a course. But if the course is not available in lecture notes form then textbook recommendations are also most welcome.

So, Can you please tell which courses in analytic number theory should I study now and give some suggestions of course pages/ textbooks? I am thinking of covering material equivalent to 2 courses.

In rest of time I am planning to study some topics from other branches of pure mathematics( one of which is Elliptic curves, is it part of analytic number theory?) depending on time available.

Should I cover more analytic number theory and less of other topics in pure mathematics?

I am badly in need of advice/ guidance on this and will be really thankful!

If you want any more detail about anything just comment.

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    $\begingroup$ Analytic number theory is fun because it is connected to everything: complex analysis, real and Fourier analysis, Galois theory, algebraic geometry and abstract algebra, groups, representations, $p$-adic numbers.. For elliptic curves & modular forms you need all of these. $\endgroup$
    – reuns
    Commented Apr 21, 2021 at 8:16
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    $\begingroup$ The list which reuns gives is a list of many places where analytic number theory interacts, but it is not necessary to learn all these to do research in analytic number theory. Still, if you want to do research in pure mathematics, it is good general advice to get as wide a background as possible since a lot of mathematics is related and having a wide viewpoint can be very useful. $\endgroup$
    – Daniel Loughran
    Commented Apr 21, 2021 at 13:47
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    $\begingroup$ In any case, given the books you have listed, the next natural text to look at would be "Iwaniec and Kowalski: Analytic Number Theory". This book gives a good modern graduate perspective on analytic number theory and will give you a very good idea of the kind of things out there. $\endgroup$
    – Daniel Loughran
    Commented Apr 21, 2021 at 13:49
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    $\begingroup$ I'm sure we can all agree on at least one thing -- we're rooting for you! I'm not a number theorist, but I have some general meta-advice: don't just talk to strangers on the internet, but seek out also people from your masters institution or others who know you personally for advice from a complementary perspective, informed by their knowledge of you personally or at least of your background. Also, what are some of your favorite theorems / topics / arguments / methods / etc. in analytic number theory? What draws you to the subject? $\endgroup$
    – Tim Campion
    Commented Apr 22, 2021 at 1:24
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    $\begingroup$ Elliptic curves are a separate branch of number theory, though of course there are overlaps. They are so ubiquitous you should definitely learn about them. $\endgroup$
    – Daniel Loughran
    Commented Apr 22, 2021 at 12:02

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(I'm posting this as an answer since I prefer to mantain my anonymity, thus I cannot comment.)

I'm also from a third-world country. At the age of 24, when I was finishing my master's thesis, I was diagnosed with bipolar disorder, and I was drinking way too much. It took me years to recover from alcoholism and to handle my depression. There were three things that kept me alive: the few people that kept around me and cared about me, teaching (part-time), and studying number theory (and that is why I'm posting this comment). Basically, it was the only thing that I was passionate about at that time. It helped me a lot to focus on positive things and to avoid depressing thoughts.

Now I'm in my 30s, and finishing my phd on number theory, and I couldn't be happier. So, dear anon, please stay positive about the future, stay away from illegal drugs and alcohol, be responsible with your treatment (I hope you find a good doctor), and keep studying what you love, in this case number theory.

More important, respect your recovery processes. It is way more important to be healthy mentally (and physically, as far as possible), than anything else.

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