Number Theory

1. Analytic Number Theory
   1. Prime Number Theorem
      1. Statement and significance
      2. Historical development
         1. Gauss's conjecture
         2. Riemann's insights
      3. Proof techniques
         1. Hadamard and de la Vallée Poussin proofs
         2. Elementary proofs and later developments
      4. Error terms and improvements
         1. Van der Corput method
         2. Vinogradov's refinement
   2. Riemann Zeta Function
      1. Definition and properties
         1. Functional equation
         2. Analytic continuation
      2. Euler product formula
      3. Non-trivial zeros
         1. Critical strip and critical line
         2. Riemann Hypothesis
            1. Statement and implications
            2. Relation to prime number distribution
         3. Computational approaches to zeros
      4. Special values of zeta function
         1. Values at even integers (e.g., ζ(2), ζ(4))
         2. Connection to Bernoulli numbers
   3. Dirichlet Characters and L-functions
      1. Introduction to characters
         1. Definition and examples
         2. Orthogonality relations
      2. Dirichlet L-functions
         1. Definition and properties
         2. Analytic continuation and functional equation
      3. Applications to prime number distribution
         1. Dirichlet's theorem on primes in arithmetic progressions
         2. Generalized Riemann Hypothesis
   4. Circle Method
      1. Introduction and historical context
      2. Hardy-Littlewood method
         1. Major arcs and minor arcs
         2. Application to Waring's problem
         3. Asymptotic formulae for representations by forms
      3. Modern developments and refinements
         1. Application to Goldbach's conjecture
         2. Applications in additive problems
   5. Sieve Methods
      1. Basic sieve methods
         1. Sieve of Eratosthenes
         2. Legendre's sieve
         3. Brun's sieve
      2. Advanced sieve techniques
         1. Selberg sieve
         2. Large sieve
         3. Applications to twin prime conjecture
      3. Applications in exponential sums
         1. Estimation of exponential sums
         2. Bombieri-Vinogradov theorem
   6. Additive Number Theory
      1. Goldbach's Conjecture
         1. Statement and historical background
         2. Partial results and contemporary approaches
      2. Waring's Problem
         1. Formulation and history
         2. Hilbert's solution
         3. Hardy and Littlewood's contributions
      3. Sumsets and structure theorems
         1. Erdős–Turán conjecture in additive problems
         2. Modern approaches and progress
   7. Expanding beyond classical topics
      1. Automorphic forms and number theory
         1. Basic introduction to automorphic forms
         2. Connections between automorphic forms and L-functions
      2. Applications to cryptography and computational problems
         1. Influence of analytic methods on cryptographic protocols
         2. Computational challenges in analytic number theory