I'm a physics graduate recently learned Riemann hypothesis in a mathematical physics course. ( I knew what the hypothesis is but didn't know mathematical statement)

I got interested, and I wanna study more about Riemann zeta function and Riemann hypothesis. I don't dare to solve it, just wanna taste it.

I've learned basic complex analysis, and calculus, linear algebra(though seems not related on this), but not number theory. Recently I borrowed a book named 'The music of primes' but I found it was too verbose even though the writer was a mathematician. (I know, the book was for laypeople)

I want mathematical approach, but I know that I have no ability to read professional papers or like that, so I would like to start with the appropriate prerequisite branch of mathematics.

Can you suggest what mathematical branch is related to the Riemann hypothesis? And if you can suggest proper textbooks which I can study by myself it would be appreciated.

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    $\begingroup$ Apostol's Introduction to Analytic Number Theory might be a good place to start. $\endgroup$ – user296602 Mar 17 '18 at 1:01
  • $\begingroup$ math.stackexchange.com/questions/153022/… is a related post too. $\endgroup$ – user296602 Mar 17 '18 at 1:01
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    $\begingroup$ I always enjoy the wikipedia page when exploring something new. Especially for the zeta function it's full of thought-candy. $\endgroup$ – vrugtehagel Mar 17 '18 at 1:03

I recognize that this question is old enough that the OP has presumably already found the book he was looking for, but in case it is helpful to others I just wanted to note that Barry Mazur and William Stein have recently published an excellent book (Prime Numbers and the Riemann Hypothesis) which aims to explain the statement and significance of the Riemann Hypothesis to readers with very minimal backgrounds. The majority of the book assumes only a basic knowledge of calculus while the end assumes a bit of complex analysis.

This past January I mentored a reading course in which a student that had just finished taking Calc 2 worked through the book. (I mentioned the Riemann hypothesis briefly during our discussion of p-series and worked out its Euler product to illustrate the connection to primes.) There were a few points that I had to explain to him, but overall my impression was that he enjoyed the book quite a bit and that it really helped him appreciate the importance of the problem. I should also point out that the book is quite inexpensive (~$20).

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Most books on analytic number theory mention the Riemann zeta function in one way or another. In fact, there is a nice Dover book (so it's fairly cheap) called Riemann's Zeta Function by Edwards: https://www.amazon.com/Riemanns-Zeta-Function-Harold-Edwards/dp/0486417409 -- that's a place to start anyway. It goes through the basics, and beyond, and only presupposes some knowledge of one variable complex analysis.

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In case you are not ware of, here is the Original statement of RH by Riemann himself .

Based on your math background (basic complex analysis, calculus, linear algebra, but not number theory) and my own experience, I would recommend the following 2 review papers:

(1) "Zeros of entire Fourier transforms" by Dimitrov and Rusev, East. J. On Approximations, vol. 17, no. 1 (2011): pp 1-108.

(2) P. M. Hallum, "Zeros of Entire Functions Represented by Fourier Transforms", Master Thesis,University of Hawaii at Manoa (2014).

The reasons that I made this recommendation are the following:

I also have a physics background and am interested in studying Riemann Hypothesis.

For some books I encountered, I found out that they contain too much advanced math (like analytical number theory) than I am familiar with, e.g., (3) "The Theory of the Riemann Zeta-Function" by Titchmarsh, (4) "Riemann Zeta Function" by Edwards and (5) "Introduction to Analytic Number Theory" by Apostol etc.

For other books I encountered, I found out that they contain too little math than I wish to see, e.g.,(6) "Prime Numbers and the Riemann Hypothesis" by Mazur and Stein, (7) "The music of primes" by Marcus du Sautoy.

But reading ref.(1) and ref.(2) does not need the advanced knowledge of analytical number theory. You only need to be familiar with Fourier transform when you start to read ref.(1) and ref.(2). So these two papers just fitted my needs and (I hope) may fit yours as well.

You can then learn other necessary math subjects as you go from math books or from wikipedia.org (like I did).

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