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