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I am planning to study number theory, and as preparation I have studied high school–level differential and integral calculus (primarily single-variable), high school algebra, and a little abstract algebra. More specifically, I am currently working through Michael Artin’s Algebra and have read the first four chapters (matrices, groups, vector spaces, and linear transformations).

In addition, I have some exposure to elementary number theory (congruences, Fermat’s little theorem, Euler’s theorem, and Wilson’s theorem), though my knowledge in this area is not extensive.

I am particularly interested in analytic number theory, since its results seem fascinating, and I hope that the proofs and underlying rigor will be equally engaging. On the other hand, the style of elementary number theory that emphasizes problem-solving in the spirit of mathematical Olympiads has not appealed to me as much.

Given this background, could you recommend a suitable book on analytic number theory that balances accessibility with rigor, and that would help me not only grasp the subject but also appreciate its depth and beauty?

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  • $\begingroup$ Possible duplicate of math.stackexchange.com/questions/153022/… $\endgroup$ Commented Sep 1 at 14:16
  • $\begingroup$ You write "the style of elementary number theory that emphasizes problem-solving in the spirit of mathematical Olympiads has not appealed to me as much." What makes a problem "in the spirit" of math Olympiads? Elementary number theory is not inherently about solving problems in that style. That you have such an impression makes me suspect you have learned elementary number theory from the wrong books. Consider Silverman's "A Friendly Introduction to Number Theory" or Stein's "Elementary Number Theory: Primes, Congruences, and Secrets". $\endgroup$ Commented Sep 3 at 5:37
  • $\begingroup$ To be honest, I have had very little formal training in elementary number theory. As a high school student in India, number theory is scarcely covered in our curriculum, and expert guidance on the subject is often hard to find. I have read a few pages from Burton, but I found Apostol more engaging, particularly since my natural inclination is toward calculus (Calculus is studied rigorously in our curriculum, and I have been fortunate to find good guidance in that area). But a genuine 'Thank you' for the suggestion! $\endgroup$ Commented Sep 3 at 6:43

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I think Apostol's Introduction to Analytic Number Theory is what you want.

https://www.amazon.com/Introduction-Analytic-Number-Theory-Apostol/dp/0387901639/ref=sr_1_1

It gives the elementary stuff then gets quite rigorous. The proof of Dirichlet's primes in arithmetic progression is worked out in detail, and that proof introduces characters, so it's the springboard for much further study.

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