# Dirichlet eta function

For $s\in \u2102$, the *Dirichlet eta function ^{}* is defined as

$$\eta (s):=\sum _{n=1}^{\mathrm{\infty}}\frac{{\left(-1\right)}^{n-1}}{{n}^{s}}.$$ | (1) |

Let $s=\sigma +it$. For $s$ a positive real number the series converges by the alternating series test^{}, by the second listed in the entry on Dirichlet series it converges for all $s$ with $\sigma >0$.

It can be shown that $\eta (s)=(1-{2}^{1-s})\zeta (s)$, where $\zeta (s)$ is the Riemann zeta function^{}. The pole of $\zeta (s)$ at $s=1$ is cancelled by the zero
of $1-{2}^{1-s}$.

Title | Dirichlet eta function |
---|---|

Canonical name | DirichletEtaFunction |

Date of creation | 2013-03-22 14:31:28 |

Last modified on | 2013-03-22 14:31:28 |

Owner | Mathprof (13753) |

Last modified by | Mathprof (13753) |

Numerical id | 9 |

Author | Mathprof (13753) |

Entry type | Definition |

Classification | msc 11M41 |

Related topic | ZerosOfDirichletEtaFunction |