AMS Chelsea Publishing 1993; 154 pp; hardcover Volume: 337 ISBN10: 0821844881 ISBN13: 9780821844885 List Price: US$30 Member Price: US$27 Order Code: CHEL/337.H
 Knopp's engaging book presents an introduction to modular functions in number theory by concentrating on two modular functions, \(\eta(\tau)\) and \(\vartheta(\tau)\), and their applications to two numbertheoretic functions, \(p(n)\) and \(r_s(n)\). They are well chosen, as at the heart of these particular applications to the treatment of these specific numbertheoretic functions lies the general theory of automorphic functions, a theory of farreaching significance with important connections to a great many fields of mathematics. The book is essentially selfcontained, assuming only a good firstyear course in analysis. The excellent exposition presents the beautiful interplay between modular forms and number theory, making the book an excellent introduction to analytic number theory for a beginning graduate student. Readership Graduate students and research mathematicians interested in analysis. Table of Contents  The Modular Group and Certain Subgroups: 1. The modular group; 2. A fundamental region for \(\Gamma(1)\); 3. Some subgroups of \(\Gamma(1)\); 4. Fundamental regions of subgroups
 Modular Functions and Forms: 1. Multiplier systems; 2. Parabolic points; 3 Fourier expansions; 4. Definitions of modular function and modular form; 5. Several important theorems
 The Modular Forms \(\eta(\tau)\) and \(\vartheta(\tau)\): 1. The function \(\eta(\tau)\); 2. Several famous identities; 3. Transformation formulas for \(\eta(\tau)\); 4. The function \(\vartheta(\tau)\)
 The Multiplier Systems \(\upsilon_{\eta}\) and \(\upsilon_{\vartheta}\): 1. Preliminaries; 2. Proof of theorem 2; 3. Proof of theorem 3
 Sums of Squares: 1. Statement of results; 2. Lipschitz summation formula; 3. The function \(\psi_s(\tau)\); 4. The expansion of \(\psi_s(\tau)\) at \(1\); 5. Proofs of theorems 2 and 3; 6. Related results
 The Order of Magnitude of \(p(n)\): 1. A simple inequality for \(p(n)\); 2. The asymptotic formula for \(p(n)\); 3. Proof of theorem 2
 The Ramanujan Congruences for \(p(n)\): 1. Statement of the congruences; 2. The functions \(\Phi_{p,r}(\tau)\) and \(h_p(\tau)\); 3. The function \(s_{p, r}(\tau)\); 4. The congruence for \(p(n)\) Modulo 11; 5. Newton's formula; 6. The modular equation for the prime 5; 7. The modular equation for the prime 7
 Proof of the Ramanujan Congruences for Powers of 5 and 7: 1. Preliminaries; 2. Application of the modular equation; 3. A digression: The Ramanujan identities for powers of the prime 5; 4. Completion of the proof for powers of 5; 5. Start of the proof for powers of 7; 6. A second digression: The Ramanujan identities for powers of the prime 7; 7. Completion of the proof for powers of 7
 Index
