From the Preface: 'An accurate (though uninspiring) title for this book would have been Applications of the Theory of the Modular Forms $\eta(\tau)$ and $\vartheta(\tau)M$ to the Number-Theoretic functions $p(n)$ and $r_s(n)$ respectively. This is accurate because, except in the first two chapters, we deal exclusively with these two modular forms and these two number-theoretic functions. However, 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'.From the Preface: 'Indeed, together with Riemann surface theory, analytic number theory has provided the principal impetus for the development over the last century of the theory of automorphic functions...I have tried to keep the book self-contained for those readers who have had a good first-year graduate course in analysis; and, in particular, I have assumed readers to be familiar with the Cauchy theory and the Lebesgue theorem of dominated convergence'.
Auflage
Sprache
Verlagsort
Verlagsgruppe
American Mathematical Society
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ISBN-13
978-0-8284-0337-5 (9780828403375)
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Schweitzer Klassifikation
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.
