DedekindZetaFunktion

The Dedekind zeta function is a fundamental object in algebraic number theory, generalizing the Riemann zeta function to number fields. For a number field K, the Dedekind zeta function is defined as a Dirichlet series: $\zeta_K(s) = \sum_{\mathfrak{a}} \frac{1}{N(\mathfrak{a})^s}$, where the sum is over all non-zero integral ideals $\mathfrak{a}$ of the ring of integers of K, and $N(\mathfrak{a})$ is the norm of the ideal $\mathfrak{a}$. The variable s is a complex number.

This series converges for complex numbers s with real part greater than 1. A key result is that the Dedekind zeta function can be analytically continued to the entire complex plane, except for a simple pole at s=1. The residue at this pole is related to important arithmetic invariants of the number field, such as its class number and regulator.

The Dedekind zeta function encodes profound information about the arithmetic structure of a number field. For example, the location and nature of its zeros are intimately connected to the distribution of prime ideals in the field. The generalized Riemann Hypothesis, which posits that all non-trivial zeros lie on the critical line with real part 1/2, is a major unsolved problem in number theory with significant implications for our understanding of number fields. The Dedekind zeta function plays a crucial role in the study of reciprocity laws and class field theory.