This is the first volume of a two volume textbook which evolved from a course (Mathematics 160) offered at the California Institute of Technology during the last 25 years. It provides an introduction to analytic number theory suitable for undergraduates with some background in advanced calculus but no previous knowledge of number theory. Actually, a great deal of the book requires no calculus at all and could profitably be studied by sophisticated high school students. Number theory is such a vast and rich field that a one[year course cannot do justice to all its parts. The choice of topics included here is intended to provide some variety and some depth. Problems which have fascinated generations of professional and amateur mathematicians are discussed together with some of the techniques for solving them. One of the goals of this course has been to nurture the intrinsic interest that many young mathematics students seem to have in number theory and to open some doors for them to the current periodical literature. It has been gratifying to note that many of the students who have taken this course during the past 25 years have become professional mathematicians, and some have made notable contributions of their own to number theory.

The Fundamental Theorem of Arithmetic / Arithmetical Functions and Dirichlet Multiplication / Averages of Arithmetical Functions / Some Elementary Theorems on the Distribution of Prime Numbers / Congruences / Finite Abelian Groups and their Characters / Dirichlet’s Theorem on Primes in Arithmetic Progressions/ Periodic Arithmetical Functions and Gauss Sums / Quadratic Residues and the Quadratic Reciprocity Law / Primitive Roots / Dirichlet Series and Euler Products / The Functions ? (s) and L(s, ?) / Analytic Proof of the Prime Number Theorem / Partitions / Bibliography / Index of Special Symbols / Index.

Undergraduate students and Teachers.