Definition:Dirichlet Eta Function

Definition

The Dirichlet $\eta$ (eta) function is the complex function defined on the half-plane $\map \Re s > 0$ as the series:

$\ds \map \eta s = \sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} n^{-s}$

Also known as

The Dirichlet $\eta$ Function is also known as the alternating $\zeta$ (zeta) function, and denoted $\map {\zeta^*} s$.

Also see

• Definition:Riemann Zeta Function
• Riemann Zeta Function in terms of Dirichlet Eta Function

• Results about the Dirichlet $\eta$ function can be found here.

Generalizations

• Definition:Dirichlet Series

Source of Name

This entry was named for Johann Peter Gustav Lejeune Dirichlet.

Sources

• Weisstein, Eric W. "Dirichlet Eta Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DirichletEtaFunction.html