Dirichlet series

A series of the form ∑ₙ₌₁∞ aₙn⁻ˢ, where the coefficients aₙ are usually complex numbers and s is a complex variable. Its convergence is governed by a half-plane and often defines an analytic function there.

Abscissa of convergence: The boundary between half-planes where a Dirichlet series converges and diverges. It summarizes the series’ convergence region with a single threshold.
Complex number: A number expressible as a real part plus an imaginary part. The variable s ranges over complex numbers, making powers n⁻ˢ complex-valued.
Riemann zeta function: The function initially defined by ζ(s)=∑ₙ₌₁∞ n⁻ˢ for real part of s greater than one, with a meromorphic continuation. It is the basic example, with every coefficient equal to one.
Power series: A series ∑ₙ aₙzⁿ in nonnegative integer powers of a variable. Its terms use powers of z, whereas Dirichlet-series terms use powers of the indices n.
Uniform convergence: Convergence in which one index threshold works simultaneously at every point in a specified domain. Uniform convergence on suitable regions justifies analytic operations on the series.
Complex exponentiation: Exponentiation extended to complex exponents using the complex exponential and a chosen logarithm. It gives the meaning of n⁻ˢ when s is not real.
Dirichlet L-function: A function defined from a Dirichlet character χ by L(s,χ)=∑ₙ₌₁∞ χ(n)n⁻ˢ in its convergence region. Character-valued coefficients create series that distinguish residue classes of primes.
Fourier series: A representation of a periodic function as a sum of sine and cosine waves or complex exponentials. Fourier series expand periodic functions in frequencies rather than encode coefficients through n⁻ˢ.
Holomorphic function: A complex-valued function that is complex differentiable throughout an open domain. Within its convergence half-plane, a Dirichlet series defines a holomorphic function.
Arithmetic function: A function defined on positive integers, commonly taking integer, real, or complex values. Many Dirichlet series encode number-theoretic data in their coefficients aₙ.

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