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Till today, I was learning "Algebra", more than other subjects (analysis/topology). I thought, learning number theory may not be difficult for me.

Many theorems/statements in number theory are easy to state, but difficult to prove, in the sense, the tools required in the proof may be from real or complex analysis.

Looking some simple statements in Number Theory, I tried to give algebraic proof, but unsuccessful. Later I came to know that proving them requires "analysis". (Dirichlet's theorem on distribution of primes, or related theorems, for example.)

If the proof of some statement is based on "algebra", I can give my effort to write the proof. However, I couldn't handle easily, right now, the tools of analysis for the problems in number theory.

Can one suggest very elementary book on analytic number theory, in which, the use of tools of analysis is illustrated with examples?

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    $\begingroup$ Introduction to Analytical Number Theory - Apostol $\endgroup$ – JVV Nov 1 '15 at 12:01
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    $\begingroup$ see this. $\endgroup$ – user795571 Nov 1 '15 at 13:08
  • $\begingroup$ @user795571: Interesting link. Thanks for the help. $\endgroup$ – Groups Nov 2 '15 at 2:53
  • $\begingroup$ you're welcome. $\endgroup$ – user795571 Nov 2 '15 at 12:40
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The first book that comes into my mind is Introduction to Analytic Number Theory by T. Apostol. It also contains some elementary number theory stuff, which is amusing and helpful as a building block.

Also take a look at Apostol's Modular Functions and Dirichlet Series in Number Theory for a little bit advanced stuff and Introduction to Number Theory by L.-K. Hua for a comprehensive overview.

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Try to look at Multiplicative Number Theory by Montgomery and Vaughan. It's written thinking at the fact that the analytic number theorists are not distributed all over the world in an homogeneus way, thus it's self contained, self explained.

Then, the subject is not easy in itself, thus even the easiest book dealing with ANT will be a little hard. However this book starts from the beginning of analytic number theory.

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