The Riemann Zeta Function
The central object of study in the Riemann Hypothesis. This remarkable function encodes deep information about prime numbers and connects analysis, number theory, and physics in unexpected ways.
Key Topics
Definition and Properties
The Riemann zeta function ζ(s) = Σ(1/n^s) for Re(s) > 1 extends meromorphically to the complex plane with a simple pole at s=1. It satisfies the functional equation relating ζ(s) to ζ(1-s), revealing deep symmetry.
Functional Equation
The functional equation π^(-s/2)Γ(s/2)ζ(s) = π^(-(1-s)/2)Γ((1-s)/2)ζ(1-s) shows the zeta function has 'functional symmetry' about the critical line Re(s) = 1/2.
Trivial and Non-trivial Zeros
The zeta function has 'trivial' zeros at negative even integers (-2, -4, -6, ...). All other zeros lie in the critical strip 0 < Re(s) < 1, and the Riemann Hypothesis claims they all lie on Re(s) = 1/2.
Relation to L-functions
The zeta function is the prototype for more general L-functions arising in number theory. These encode arithmetic information about algebraic varieties, modular forms, and number fields.
Recent Papers
Computational Verification of Zeta Zeros
Dr. Lisa Wang, Prof. Alexandre Petrov • 2024
L-functions and Their Zeros
Dr. Robert Müller • 2023
The Functional Equation and Its Consequences
Prof. Alexandre Petrov • 2023