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Modular Functions in Analytic Number Theory: Second Edition
Marvin I. Knopp, Temple University, Philadelphia, PA

AMS Chelsea Publishing
1993; 154 pp; hardcover
Volume: 337
ISBN-10: 0-8218-4488-1
ISBN-13: 978-0-8218-4488-5
List Price: US$30
Member Price: US$27
Order Code: CHEL/337.H
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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 number-theoretic 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 number-theoretic functions lies the general theory of automorphic functions, a theory of far-reaching significance with important connections to a great many fields of mathematics.

The book is essentially self-contained, assuming only a good first-year 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.


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
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