ZetaFunktion

The zetafunktion, commonly called the Riemann zeta function, is a complex function ζ(s). For Re(s) > 1, it is defined by the Dirichlet series ζ(s) = sum_{n=1}^∞ 1/n^s. It extends meromorphically to the entire complex plane, with a single simple pole at s = 1.

For Re(s) > 1 the function also has the Euler product representation ζ(s) = product over primes p of (1 − p^(-s))^(-1). This product encodes the distribution of prime numbers into the analytic properties of the function.

The zetafunktion satisfies a functional equation that relates ζ(s) to ζ(1−s) through factors involving the gamma function and powers of π. It has trivial zeros at negative even integers s = −2, −4, … and nontrivial zeros in the critical strip 0 < Re(s) < 1; the famous Riemann hypothesis asserts that all nontrivial zeros lie on the critical line Re(s) = 1/2. Special values: for positive integers n, the even values ζ(2n) can be expressed in terms of Bernoulli numbers and powers of π.

Significance and generalizations: ζ(s) is central to analytic number theory and the study of prime distribution, underpinning results such as the Prime Number Theorem. Generalizations include the Hurwitz zeta function ζ(s, a) and Dirichlet L-functions; the zetafunktion is the prototypical example from which many other zeta-type functions are developed. In physics and mathematics, zeta-type functions appear in regularization techniques and spectral theory.