Dirichlet eta function
📂FunctionsDirichlet eta function
Definition
The function η:C→C defined as below is called the Dirichlet eta Function.
η(s):=n∈N∑(−1)n−1n−s
The Dirichlet eta function is defined as the alternating Riemann zeta function.
Theorems
- [1] Relationship with the Riemann zeta function:
η(s)=(1−21−s)ζ(s)
- [2] Relationship with the gamma function: If Re(s)>1, then
η(s)Γ(s)=M[ex+11](s)=∫0∞ex+1xs−1dx
- Re(z) represents the real part of the complex number z∈C.
Proof
[1]
ζ(s)−η(s)=====n∈N∑ns1−n∈N∑ns(−1)n−1n∈N∑(ns1+ns(−1)n)2n∈N∑(2n)s121−sn∈N∑ns121−sζ(s)
Modifying for the Dirichlet eta function gives
η(s)=(1−21−s)ζ(s)
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[2]
It is not so simple. Derived using the Dominated Convergence Theorem.
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