Dirichlet eta function

For s, the Dirichlet eta functionMathworldPlanetmath is defined as

η(s):=n=1(-1)n-1ns. (1)

Let s=σ+it. For s a positive real number the series converges by the alternating series testMathworldPlanetmath, by the second listed in the entry on Dirichlet series it converges for all s with σ>0.

It can be shown that η(s)=(1-21-s)ζ(s), where ζ(s) is the Riemann zeta functionDlmfDlmfMathworldPlanetmath. The pole of ζ(s) at s=1 is cancelled by the zero of 1-21-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