Introduction to the Riemann Hypothesis and Analytic Number Theory

The Riemann Hypothesis stands as one of the most profound and fascinating challenges in modern mathematics, lying at the very heart of prime number theory. Understanding this hypothesis requires a clear journey through complex analysis and arithmetic functions. This text-based course guides you step-by-step through the core mathematical machinery behind the hypothesis, making advanced analytic number theory accessible to curious minds. You will transition from grasping basic complex variables to understanding how the distribution of prime numbers is intimately connected to the zeros of the Riemann Zeta function. By reading through structured proofs, clear mathematical explanations, and illustrative derivations, you will build a strong intuitive and formal foundation in this beautiful branch of mathematics. What you'll learn: - Understand the foundational definitions of the Riemann Zeta function and its domain of convergence. - Explore the connection between prime numbers and the Euler product formula. - Analyze the properties of holomorphic and meromorphic functions in the complex plane. - Study the analytic continuation of the Zeta function and its functional equation. - Examine the Prime Number Theorem and how the non-vanishing of the Zeta function on the 1-line relates to it. - Investigate the formulation of the Riemann Hypothesis and its modern applications in cryptography and computation. This course begins with essential terminology, complex analysis prerequisites, and foundational definitions before moving into the core properties of the Zeta function and its applications. You will follow a logical progression of written proofs, detailed mathematical commentary, and conceptual summaries designed to solidify your understanding. This course is designed for undergraduate students, mathematics enthusiasts, and beginners to analytic number theory who have a basic background in calculus and algebra. No prior knowledge of advanced complex analysis is required. Begin your journey into the deep connections between complex analysis and prime numbers today.