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I have general question on Riemann Zeta function. How can I improve knowledge on Riemann Zeta Function theory up to research? For example , what are the best books on Zeta Function theory? I wish to learn this theory as deep as possible... Thanks

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H.M. Edwards' book is wonderful IMO. –  Ron Gordon Jan 29 at 18:55
    
Classical masterpiece is "The Theory of Riemann Zeta Function" by Titchmarsh –  i707107 Jan 29 at 18:57
    
I found this book good "Apostol, Tom M. Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York." –  Maryam Jan 29 at 19:10

2 Answers 2

I think that it could be even more reasonable to read first a good introduction to number theory in general. This usually has a chapter (or more) on the Riemann $\zeta$-function. Of course, one could argue why a book on number theory in general, and not about analytic number theory in particular, or even only on Dirichlet series. However, if you want to come "up to research", this view is too narrow, I think. There are so many wonderful topics in number theory which involve L-series and Zeta-functions, that it would be a pity not to know about it, and only struggle with zero-free regions for $\zeta(s)$, exponential sums etc.

My suggestion: Ireland, Rosen: A classical introduction to modern number theory. Chapter 11: The Zeta function, Chapter 16: Dirichlet L-function.

If this is too elementary for you, then there are more specific books (Edward C. Titchmarsh, Aleksandar Ivic, W. Narkievic, etc.)

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An outstanding starting point is Stopple "Primer of Analytic Number Theory." Great presentation of RZF, Bernoulli numbers, and motivating problems that compliment the material and enhance intuitive understanding - plus a complete set of solutions.

As recommended above, Apostol.

For specifics about the RZF, a classic is Edwards "Riemann Zeta Function."

Then continuing along the lines suggested above, an excellent book on zeta functions and other topics in number theory consider Kato "Fermat's Dream."

Then you could also go to Montgomery "Multiplicative Number Theory" which has numerous pertinent sections.

And then to Ivic (sorry for the missing accent) "Riemann Zeta Function"

Also, not to be overlooked are the numerous questions and often extremely detailed answers here. And, of course, at mathoverflow.net

Not being at the level of your intended destination, I am sure there are many additional sources that will also benefit you.

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