Question: What would be some must-read papers for an aspiring analytic number theorist? In other words, what are the papers that any analytic number theorist would have read? (Background: Someone who has seen some classical analytic number theory, at the level of Davenport and modular forms books by Iwaniec)

There are some classical papers that comes to mind. For example,

  • Duke-Friedlander-Iwaniec's work on subconvexity bounds
  • Iwaniec's 1987 paper on bounding Fourier coefficients of half-integral weight modular forms
  • Duke's work on Linnik's problems

Basically I would like to collect papers that any graduate student in analytic number theory should have read/should be aware of. In particular I am interested in something more modern (although very classical papers such as Riemann's memoir is still welcome, for the sake of completeness) Thank you.

Edit: In light of Professor Garrett's comment, I am trying to restrict the scope a little bit. (Hopefully it's now more reasonable) Analytic number theory is a pretty diverse area, but hopefully there is a reasonable list of papers that every analytic number theorist would have read/know about.

  • 4
    $\begingroup$ There are probably at least 100, if not a large factor more... if only because "analytic number theory" has many different senses, and many different corresponding notions of "background". Surely a list of 1000 papers would not be a desired answer... Clarification is needed, or else some sort of acknowledgement of "oops, so there's no royal road..."? $\endgroup$ Feb 27, 2013 at 23:14
  • 7
    $\begingroup$ I feel like Iwaniec and Kowalski's "Analytic Number Theory" is a great guide to the further literature. I would suggest to an aspiring analytic number theorist that he or she first read Davenport, then start reading parts of IK, then follow up on whichever parts of IK seem most interesting. "Opera de Cribro" by Friedlander and Iwaniec is another great book that discusses much of the very recent literature. Also, Soundararajan has some excellent survey papers (such as "The Distribution of the Primes") with more references. There's much more out there, I am interested in other responses too. $\endgroup$ Feb 28, 2013 at 3:18
  • 1
    $\begingroup$ I'd venture to bet that now with your restriction, taken in a strict sense, you are asking for the empty-set. I doubt there is even a single paper everybody (or even almost everybody) that qualifies as analytic number theorist has read. On a more serious note: what books have you already read? $\endgroup$
    – user9072
    Feb 28, 2013 at 10:24
  • 5
    $\begingroup$ This question is too broad and your background is unclear. I am not an analytic number theorist, but I'll throw in my two cents. Weil's "explicit formula" paper and the paper where he proves the converse to Hecke's lemma. Selberg's collected works. $\endgroup$ Feb 28, 2013 at 12:35
  • 1
    $\begingroup$ I think that most analytic number theorists have at least a passing familiarity with the entire contents of IK, and know a little bit about sieve methods, but most of the successful young analytic number theorists I know started doing research before reading a lot of the literature. However, I think Iwaniec encourages his students to read very heavily in grad school, and several of them are extremely successful; you would certainly benefit by mastering any of his books. $\endgroup$ Feb 28, 2013 at 14:46


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.