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Context: I am a junior math major and am hoping to go to grad school after next year for a PhD. I have completed most of the standard undergraduate courses and have been consistently most interested by number theory. I've taken three courses in number theory; the first using Niven's text, the second using Ireland and Rosen, and the third using Montgomery and vaughan's Multiplicative Number Theory.

Next year, I have room for four independent study courses. One of them will likely be on (introductory) algebraic geometry but with the other three, I wish to gain some breadth within number theory to find out what I am most passionate about.

Questions: What are the popular branches of number theory being actively researched today and what are some good introductory texts in each of these?

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migrated from mathoverflow.net Apr 27 '14 at 3:48

This question came from our site for professional mathematicians.

  • $\begingroup$ I'd say all of them to the question's title. $\endgroup$ – DonAntonio Apr 27 '14 at 3:53
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    $\begingroup$ Dear user, Have you looked at this question and its answers? Regards, $\endgroup$ – Matt E Apr 27 '14 at 4:44
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Gaps between primes is certainly active at the moment so I'd recommend Harman's book Prime-Detecting Sieves. Generally Iwaniec and Kowalski's book Analytic Number Theory and Hardy & Wright's book Introduction to the Theory of Numbers. Also I'd say that the circle method is fairly commonly used at the moment and Bob Vaughan has a great book on it entitled the Hardy-Littlewood Method.

Also sumset problems seem to be big at the moment and Tao & Vu's book Additive Combinatorics is a great read! Ben Green has some good expository papers online on the topic as well.

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  • $\begingroup$ Thank you very much for your speedy response! I am looking into the suggested texts on google books now and will definitely head over to the library tomorrow. $\endgroup$ – user50065 Apr 27 '14 at 2:42
  • $\begingroup$ Sorry, I made a slight mistake, it should be the Iwaniec/Kowalski book. I had the Rosser-Iwaniec sieve on my mind. Whoops. $\endgroup$ – Stijn Hanson Apr 27 '14 at 2:45
  • $\begingroup$ I've paged through it it in the library before so I figured, but worry that it would overlap too much with Montgomery and Vaughan's for my department to approve an independent study using it. (Maybe not, do they overlap less than I recall?) $\endgroup$ – user50065 Apr 27 '14 at 2:53
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    $\begingroup$ Analytic number theory is a big area, I don't think they overlap too much but I've not really done an intensive study into Iwaniec and Kowalski, just flicked through it a little bit and heard many great things. Rather interestingly, I was talking with Vaughan the other week and the next Multiplicative Number Theory book should be finished next year, hopefully. $\endgroup$ – Stijn Hanson Apr 27 '14 at 3:00
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    $\begingroup$ Also, if you haven't already, check out Terry Tao's blog as that's got a lot of good exposition on a broad array of topics. $\endgroup$ – Stijn Hanson Apr 27 '14 at 3:01

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