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Modular Functions in Analytic Number Theory

Marvin I. Knopp
Publisher: 
AMS/Chelsea Publishing
Publication Date: 
2003
Number of Pages: 
154
Format: 
Hardcover
Edition: 
2
Price: 
29.00
ISBN: 
978-0828403375
Category: 
Monograph
BLL Rating: 

The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

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  • 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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