Number Theory

Below is a list of reviews of books on number theory.

An Introduction to the Theory of Numbers
Niven, I., Zuckerman, H. S., Montgomery, H. L. (5th Ed. 1991). ISBN 9780471625469
Prerequisites: minimally, calculus and comfort with proofs; but it helps to know linear and abstract algebra

Kind of an old fashioned book, but it does the job well. The first few chapters are basically what one might see in an introductory course in number theory, and then the later chapters might follow in successive courses or even intro grad level courses, minus the use of heavy abstract algebra. This textbook is a decent way to introduce elementary number theory without the use of abstract algebra, and for that reason it could be valuable to take a look at prior to a first course in abstract algebra or concurrently with one.

An Introduction to Analytic Number Theory
Apostol, T. (1976). ISBN 9781441928054
Single variable calculus minimally for the first half, complex analysis (at the level of mastery of the Cauchy integral formula) needed for the second half.

Contains most topics from a first course in elementary number theory, together with several topics from analytic number theory, including growth estimates for basic arithmetic functions, Dirichlet’s theorem on primes in arithmetic progressions, and the prime number theory, with some basic idea from the theory of partitions at the end. The book contains a miraculous amount of information in one place, and manages to minimize the prereqs needed to an astounding degree. In particular, while almost all courses in analytic number theory would need complex analysis at a seriously high level as a prereq, one could run a course from this book with complex analysis as a coreq. However, this comes at a price: many proofs don’t truly show the student what’s going on. In particular, nonvanishing of Dirichlet L-functions at 1 is typically proven by pole-zero cancellation (in the complex character case) and either by miraculous complex analysis or by the class number formula (in the real character case); Apostol manages to use pole-zero cancellation type ideas without recourse to complex analysis to handle the complex character case, and summation along hyperbolas in the real character case, leading to elementary proofs that are less well motivated. Overall, this is a great resource for a course in complex analysis, but a bit harder to learn from via self study. A serious student of analytic number theory would also benefit from studying more advanced treatments (such as Davenport) after working through this text.