Guidance on classes I should take?

[dh363]

Hey guys, I'm a second semester sophomore who goes to a school that's not terribly well known for math. As such, I'll have taken most of the math classes I want to take by senior year (we don't have a very strong graduate program, so not a lot of good grad courses except for applied math/stats). By the end of junior year, I plan to have taken:

Multivariable Calculus
Linear Algebra
A proofs and problem solving class
Grad level Probability Theory and Applications
Abstract Algebra
Analysis I
Analysis II
Complex Analysis
ODEs
PDEs
Differential Geometry

I have around 4 spaces for more math classes senior year. I've taught myself analysis out of GF Simmons' book and number theory from Hardy and Wright. I'm very interested in going into analytic number theory for grad school. We have a few profs here whose research focuses on things from analysis, several complex variables, number theory, PDEs, etc, and I was thinking of asking some profs I know well to do independent studies with me, which the school allows with approval from the faculty member. I was wondering what sort of things I should learn to best position myself for studying analytic number theory in grad school

Thanks

 

[micromass]

Hmm, wouldn't it be best to actually take a book on analytic number theory and try to understand the material? You will likely not understand everything in the book, so you will have to read up on number theory and analysis.

Another thing you should know more about is complex analysis. You already took a course, but I bet you didn't come to the interesting parts yet. With complex analysis, it is for example possible to give a proof of the celebrated prime number theorem. Why don't you find a complex analysis textbook which contains that and study it? I know that Freitag & Busam contains a discussion and proof on the prime number theorem, and there are certainly other textbooks that do.

There are a lot of powerful methods used in analytic number theory. One of these methods is the so-called tauberian theory. Perhaps you could study that? Other techniques are "sieve methods".

Multiple complex variables is also used in modern analytic number theory, so it won't hurt to know something about that.

Probability theory is another thing you might study. A very recent development is in the study of number theory through the tools of probability. Maybe this can be of interest of you??

 

[dh363]

Hmm, wouldn't it be best to actually take a book on analytic number theory and try to understand the material? You will likely not understand everything in the book, so you will have to read up on number theory and analysis.

Another thing you should know more about is complex analysis. You already took a course, but I bet you didn't come to the interesting parts yet. With complex analysis, it is for example possible to give a proof of the celebrated prime number theorem. Why don't you find a complex analysis textbook which contains that and study it? I know that Freitag & Busam contains a discussion and proof on the prime number theorem, and there are certainly other textbooks that do.

There are a lot of powerful methods used in analytic number theory. One of these methods is the so-called tauberian theory. Perhaps you could study that? Other techniques are "sieve methods".

Multiple complex variables is also used in modern analytic number theory, so it won't hurt to know something about that.

Probability theory is another thing you might study. A very recent development is in the study of number theory through the tools of probability. Maybe this can be of interest of you??

I am planning on studying some number theory via Hardy & Wright, then try to learn some more analysis before moving onto Apostol's introduction to anlaytic number theory. Would this be too hard or a good progression?

I'm also thinking of approaching my complex analysis professor, who seems to like me a lot, and asking to do an independent study with him in more advanced complex analysis and then multiple complex variables (one of his research areas) the semester after that.