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Dirichlet eta function 📂Functions

Dirichlet eta function

Definition

The function η:CC\eta : \mathbb{C} \to \mathbb{C} defined as below is called the Dirichlet eta Function. η(s):=nN(1)n1ns \eta (s) := \sum_{n \in \mathbb{N}} (-1)^{n-1} n^{-s}

The Dirichlet eta function is defined as the alternating Riemann zeta function.

Theorems

  • [1] Relationship with the Riemann zeta function: η(s)=(121s)ζ(s) \eta (s) = \left( 1 - 2^{1-s} \right) \zeta (s)
  • [2] Relationship with the gamma function: If Re(s)>1\operatorname{Re} (s) > 1, then η(s)Γ(s)=M[1ex+1](s)=0xs1ex+1dx \eta (s) \Gamma (s) = \mathcal{M} \left[ {{ 1 } \over { e^{x} + 1 }} \right] (s) = \int_{0}^{\infty} {{ x^{s-1} } \over { e^{x} + 1 }} dx

  • Re(z)\Re(z) represents the real part of the complex number zCz \in \mathbb{C}.

Proof

[1]

ζ(s)η(s)=nN1nsnN(1)n1ns=nN(1ns+(1)nns)=2nN1(2n)s=21snN1ns=21sζ(s) \begin{align*} \zeta (s) - \eta (s) =& \sum_{n \in \mathbb{N}} {{ 1 } \over { n^{s} }} - \sum_{n \in \mathbb{N}} {{ (-1)^{n-1} } \over { n^{s} }} \\ =& \sum_{n \in \mathbb{N}} \left( {{ 1 } \over { n^{s} }} + {{ (-1)^{n} } \over { n^{s} }} \right) \\ =& 2 \sum_{n \in \mathbb{N}} {{ 1 } \over { (2n)^{s} }} \\ =& 2^{1-s} \sum_{n \in \mathbb{N}} {{ 1 } \over { n^{s} }} \\ =& 2^{1-s} \zeta (s) \end{align*} Modifying for the Dirichlet eta function gives η(s)=(121s)ζ(s) \eta (s) = \left( 1 - 2^{1-s} \right) \zeta (s)

[2]

It is not so simple. Derived using the Dominated Convergence Theorem.