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Modular Functions in Analytic Number Theory
Seiten
2002
|
2nd Revised edition
American Mathematical Society (Verlag)
978-0-8218-4488-5 (ISBN)
American Mathematical Society (Verlag)
978-0-8218-4488-5 (ISBN)
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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. 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. (CHEL/337.H)
Erscheint lt. Verlag | 1.1.2008 |
---|---|
Reihe/Serie | AMS Chelsea Publishing |
Verlagsort | Providence |
Sprache | englisch |
Gewicht | 393 g |
Themenwelt | Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie |
ISBN-10 | 0-8218-4488-1 / 0821844881 |
ISBN-13 | 978-0-8218-4488-5 / 9780821844885 |
Zustand | Neuware |
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