Definition & Significance
In mathematics, a zeta function is a special type of complex-valued function that generalizes the Riemann zeta function. These functions play a foundational role in analytic number theory, algebraic geometry, and theoretical physics. The prototypical example, introduced by Leonhard Euler and later studied deeply by Bernhard Riemann, is defined for complex numbers \(s\) with real part greater than 1 by the Dirichlet series:
Through analytic continuation, ζ(s) can be extended to the entire complex plane except for a simple pole at \(s = 1\). Its non-trivial zeros are conjectured to lie on the critical line \(\Re(s) = 1/2\), a statement known as the Riemann Hypothesis, one of the most famous unsolved problems in mathematics.
| Domain | Complex plane ℂ \ {1} (via analytic continuation) |
|---|---|
| Key Property | Euler product: ζ(s) = ∏p prime (1 - p-s)-1 |
| Functional Equation | ζ(s) = 2sπs-1 sin(πs/2) Γ(1-s) ζ(1-s) |
| Critical Line | Re(s) = 1/2 |
| Applications | Prime distribution, string theory, quantum chaos, statistical mechanics |