# Dirichlet eta function

For $s\in\mathbb{C}$, the Dirichlet eta function is defined as

 $\eta(s):=\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n-1}}{n^{s}}\,.$ (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 DirichletEtaFunction 2013-03-22 14:31:28 2013-03-22 14:31:28 Mathprof (13753) Mathprof (13753) 9 Mathprof (13753) Definition msc 11M41 ZerosOfDirichletEtaFunction