A primer of analytic number theory : from Pythagoras to the Riemann hypothesis

A little background on me. I have just finished my freshman year of high school, and this was my first book on number theory. However, I have read many other math texts. In the beginning of the book there are some new concepts introduced, but they are not too hard to understand. The middle is refreshing as it involves a lot of calculus, which the student is most likely familiar with. The latter part consists of a variety of new ideas, and the theorems can get quite lengthy. I do not fully understand all of them myself. The book is well written and also includes the history of many mathematical problems. This 2003 undergraduate introduction to analytic number theory develops analytic skills in the course of studying ancient questions on polygonal numbers, perfect numbers and amicable pairs. The question of how the primes are distributed amongst all the integers is central in analytic number theory. This distribution is determined by the Riemann zeta function, and Riemann's work shows how it is connected to the zeroes of his function, and the significance of the Riemann Hypothesis. Starting from a traditional calculus course and assuming no complex analysis, the author develops the basic ideas of elementary number theory. The text is supplemented by series of exercises to further develop the concepts, and includes brief sketches of more advanced ideas, to present contemporary research problems at a level suitable for undergraduates. In addition to proofs, both rigorous and heuristic, the book includes extensive graphics and tables to make analytic concepts as concrete as possible. "This undergraduate introduction to analytic number theory develops analytic skills in the course of a study of ancient questions on polygonal numbers, perfect numbers, and amicable pairs. The question of how the primes are distributed among all integers is central in analytic number theory. This distribution is determined by the Riemann zeta function, and Riemann's work shows how it is connected to the zeros of his function and the significance of the Riemann Hypothesis."--Jacket This 2003 undergraduate-level introduction describes those mathematical properties of prime numbers that can be deduced with the tools of calculus. The capstone of the book is a brief presentation of the Riemann zeta function and of the significance of the Riemann Hypothesis.

an Undergraduate-level Introduction Whose Only Prerequisite Is A Standard Calculus Course.

The Greek word gnomon means the pointer on a sundial, and also a carpenter's square or L-shaped bar.