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

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.

• 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
• 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