Modular Functions in Analytic Number Theory

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.