Edit: I have asked this question separately because I am in need of some resources which focus on analytic number theory only.So, this question is not a duplicate of:Resources where I can find open problems in number theory along with their level of difficulty as I am in looking for websites/books/blogs in Analytic Number Theory only. I don't have much knowledge of other branches of number theory.

I have completed my master's in mathematics a couple of years ago from a good university of my country but due to very strong personal and professional reasons I couldn't get admitted to grad school despite having a good academic background.

I have a really good background in analytic number theory. I have studied 1 course in Sieve Theory, 2 courses which were based on 2 volumes of Apostol's Number Theory books. I will start a course in Automorphic Forms soon. I don't have guidance of any professor right now and I want to try working on an open problem in Analytic Number Theory.

Can you please let me know of resources( websites/ blogs/books) where I can find open problems in analytic number theory to work on? Since, I don't have a guidence of a prof. due to various reasons, these resources will really help me.

Thank you!

  • $\begingroup$ it would probably be best to get in contact with some professor/postdoc somewhere. In the meantime, you could just look at recent arxiv papers in NT and try to improve some of the results you like :) $\endgroup$ Jul 11, 2022 at 9:11
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    $\begingroup$ Didn't you already ask that question? Resources where I can find open problems in number theory along with their level of difficulty $\endgroup$ Jul 11, 2022 at 13:54
  • $\begingroup$ @CarloBeenakker Here, I am interested in problems related to analytic number theory only. Sorry for the inconvinience. $\endgroup$
    – Arnold
    Jul 11, 2022 at 14:47
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    $\begingroup$ When I was a student, a nice open problem in analytic number theory was to find the optimal constant in the large sieve inequality ($1/\delta+N-1$; if you do not know what this means, look at a book explaining the large sieve). It now has several proofs, but without reading them you can try your hand at that problem. $\endgroup$ Jul 11, 2022 at 15:18
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    $\begingroup$ Try reading "Analytic Number Theory" by Iwaniec and Kowalski; littered throughout are open questions. $\endgroup$ Jul 11, 2022 at 16:53


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