RiemannZetaFunktion

The Riemannzetafunktion, more commonly known as the Riemann zeta function, is a complex function defined for complex numbers s with real part greater than one by the absolutely convergent series \(\zeta(s)=\sum_{n=1}^{\infty} n^{-s}\). It extends analytically to all complex numbers except for a simple pole at \(s=1\) with residue 1. The function plays a central role in analytic number theory, particularly in the study of the distribution of prime numbers.

Euler discovered that for positive even integers the values of the zeta function can be expressed in terms of powers of π and the Bernoulli numbers: \(\zeta(2k)=(-1)^{k+1}\frac{(2\pi)^{2k}B_{2k}}{2(2k)!}\). For negative integers the function vanishes at all negative even integers, a fact known as the trivial zeros, while the nontrivial zeros lie in the critical strip \(0<\Re(s)<1\). The famous Riemann hypothesis conjectures that all nontrivial zeros have real part equal to one-half.

The zeta function satisfies a functional equation relating its values at s and 1–s: \(\zeta(s)=2^s\pi^{s-1}\sin(\frac{\pi s}{2})\Gamma(1-s)\zeta(1-s)\). This symmetry underpins many of its analytic properties and links the function to other special functions such as the gamma function.

Applications of the Riemannzetafunktion extend beyond prime number theory. It appears in quantum physics, statistical mechanics, and in the theory of random matrices. In number theory it is used to formulate and study Dirichlet L‑functions, modular forms, and various generalizations such as the Hurwitz zeta function and multiple zeta values. The zeta function thus remains a fundamental example of an analytic object whose study bridges diverse areas of mathematics and physics.