zetafunction

Zetafunction, usually referred to as the zeta function, is a central object in analytic number theory and complex analysis. In its most common form, the Riemann zeta function, denoted zeta(s), is defined for complex numbers s with real part greater than 1 by the convergent series zeta(s) = sum_{n=1}^{∞} n^{-s}. It admits analytic continuation to a meromorphic function on the entire complex plane, with a single simple pole at s = 1.

An important alternate expression is the Euler product over primes: zeta(s) = product over all primes p of (1 - p^{-s})^{-1}, valid for Re(s) > 1. This product encodes the distribution of primes and connects the zeta function to prime numbers.

The zeta function satisfies a functional equation that relates its values at s and 1−s, allowing continuation to the whole plane. A common form is zeta(s) = 2^s pi^{s-1} sin(pi s/2) Gamma(1−s) zeta(1−s). The function has zeros in the critical strip 0 < Re(s) < 1, called nontrivial zeros; trivial zeros occur at negative even integers s = −2, −4, −6, …. The Riemann Hypothesis posits that all nontrivial zeros lie on the critical line Re(s) = 1/2.

Special values include zeta(2) = π^2/6, zeta(0) = −1/2, and zeta(−1) = −1/12. More generally, zeta(−n) = (−1)^n B_{n+1}/(n+1) in terms of Bernoulli numbers.

Generalizations of the zeta function include the Hurwitz zeta function, Dirichlet L-functions, and Dedekind zeta functions of number fields, which satisfy analogous analytic properties and functional equations.

Applications span the distribution of primes via explicit formulas, the prime number theorem, and appearances in mathematical physics and probability theory.