The Riemann zeta function is a cornerstone of analytic number theory. It's defined as an infinite series that converges for complex numbers with real part greater than 1. This function extends to the entire complex plane, except for a simple pole at s = 1.

The zeta function connects deeply to prime numbers through the Euler product formula. It's crucial in the Riemann Hypothesis and has applications in cryptography and quantum physics. Understanding its properties and behavior is key to unlocking mathematical mysteries.

Definition and Basic Properties

Defining the Riemann Zeta Function

• Riemann zeta function denoted as ζ(s) where s is a complex number
• Expressed as an infinite series: $ζ(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$
• Dirichlet series representation of the zeta function
• Converges absolutely for all complex numbers with real part greater than 1
• Extends analytically to the entire complex plane except for a simple pole at s = 1

Convergence and the Critical Strip

• Absolute convergence occurs when Re(s) > 1
• Critical strip defined as the region where 0 ≤ Re(s) ≤ 1
• Conditional convergence within the critical strip, except at s = 1
• Importance of the critical strip in the study of prime numbers and the distribution of zeros

Properties and Significance

• Fundamental object in analytic number theory
• Connects to prime number distribution through the Euler product formula
• Plays a crucial role in the Riemann Hypothesis, one of the most famous unsolved problems in mathematics
• Used in various areas of mathematics (cryptography, quantum physics)

Euler Product and Functional Equation

Euler Product Formula

• Euler product formula expresses the zeta function as a product over prime numbers
• Formula: $ζ(s) = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}}$
• Valid for Re(s) > 1
• Demonstrates the deep connection between the zeta function and prime numbers
• Provides a bridge between additive and multiplicative number theory

The Functional Equation

• Relates values of ζ(s) to values of ζ(1-s)
• Symmetric form: $π^{-s/2} Γ(s/2) ζ(s) = π^{-(1-s)/2} Γ((1-s)/2) ζ(1-s)$
• Γ(s) represents the gamma function
• Allows for the analytic continuation of ζ(s) to the entire complex plane
• Reveals the symmetry of the zeta function about the critical line Re(s) = 1/2

Zeros of the Zeta Function

Trivial Zeros

• Occur at negative even integers: s = -2, -4, -6, ...
• Result from the poles of the gamma function in the functional equation
• Easily predictable and less interesting from a number-theoretic perspective
• Do not contribute to the deeper mysteries surrounding the zeta function

Non-trivial Zeros

• Located within the critical strip 0 < Re(s) < 1
• Symmetric about the real axis and the critical line Re(s) = 1/2
• Riemann Hypothesis conjectures that all non-trivial zeros lie on the critical line
• Distribution of non-trivial zeros closely related to the distribution of prime numbers
• Studying these zeros provides insights into prime number theory and other areas of mathematics