This is the second edition of a two-volume textbook on number theory, which evolved from a course offered at the California Institute of Technology over the past 25 years. The second volume assumes a background in number theory similar to that provided in the first volume, along with a basic understanding of complex analysis. The main focus is on elliptic functions and modular functions, including their number-theoretic applications. Key topics include Rademacher's convergent series for the partition function, Lehner's congruences for the Fourier coefficients of the modular function $ j(\tau) $, and Hecke's theory of entire forms with multiplicative Fourier coefficients. The final chapter discusses Bohr's theory of equivalence of general Dirichlet series. Both volumes emphasize classical aspects of number theory, which has undergone significant modern development. The book aims to help nonspecialists become familiar with an important and fascinating part of mathematics while providing background knowledge for specialists. The second edition includes an alternate treatment of the transformation formula for the Dedekind eta function, with a five-page supplement to Chapter 3. Other changes include corrections of misprints, minor adjustments in the exercises, and an updated bibliography. The book is divided into eight chapters, covering topics such as elliptic functions, modular functions, the Dedekind eta function, congruences for the coefficients of the modular function $ j $, Rademacher's series for the partition function, modular forms with multiplicative coefficients, Kronecker's theorem, and general Dirichlet series. Each chapter includes exercises and a supplement to Chapter 3. The book also features a bibliography, an index of special symbols, and an index.This is the second edition of a two-volume textbook on number theory, which evolved from a course offered at the California Institute of Technology over the past 25 years. The second volume assumes a background in number theory similar to that provided in the first volume, along with a basic understanding of complex analysis. The main focus is on elliptic functions and modular functions, including their number-theoretic applications. Key topics include Rademacher's convergent series for the partition function, Lehner's congruences for the Fourier coefficients of the modular function $ j(\tau) $, and Hecke's theory of entire forms with multiplicative Fourier coefficients. The final chapter discusses Bohr's theory of equivalence of general Dirichlet series. Both volumes emphasize classical aspects of number theory, which has undergone significant modern development. The book aims to help nonspecialists become familiar with an important and fascinating part of mathematics while providing background knowledge for specialists. The second edition includes an alternate treatment of the transformation formula for the Dedekind eta function, with a five-page supplement to Chapter 3. Other changes include corrections of misprints, minor adjustments in the exercises, and an updated bibliography. The book is divided into eight chapters, covering topics such as elliptic functions, modular functions, the Dedekind eta function, congruences for the coefficients of the modular function $ j $, Rademacher's series for the partition function, modular forms with multiplicative coefficients, Kronecker's theorem, and general Dirichlet series. Each chapter includes exercises and a supplement to Chapter 3. The book also features a bibliography, an index of special symbols, and an index.