DirichletReihe

Dirichletreihe, named after the German mathematician Peter Gustav Lejeune Dirichlet, is a type of infinite series used primarily in number theory and complex analysis. It is generally expressed in the form:

D(s) = ∑_{n=1}^∞ a(n) n^{-s}

where a(n) is a complex-valued arithmetic function, and s is a complex variable typically written as s = σ + it. The series converges for certain values of σ, depending on the properties of the coefficients a(n).

A prominent example of a Dirichletreihe is the Riemann zeta function ζ(s), which corresponds to the case where a(n) = 1 for all n. This series converges when the real part of s, σ, exceeds 1, and can be analytically continued to a meromorphic function defined over the entire complex plane, except for a simple pole at s = 1.

Dirichletreihe are fundamental in studying properties of prime numbers, as they encode arithmetic information through their coefficients. The Euler product, which represents these series as an infinite product over primes, links the structure of primes directly to the analytic properties of the series.

The convergence, analytic continuation, and functional equations of Dirichletreihe are central topics in modern number theory and complex analysis, especially in understanding the distribution of primes. Their applications extend to L-functions, automorphic forms, and various areas in modern mathematics that involve deep relationships between algebraic and analytical structures.