Binomial Form of Relation between Riemann Zeta Function and Dirichlet Eta Function

Theorem

Let $\zeta$ be the Riemann zeta function.

Let $s \in \C$ be a complex number with real part $\map \Re s > 1$.

Then: $\ds \map \zeta s = \frac 1 {1 - 2^{1 - s} } \sum_{n \mathop = 0}^\infty \paren {\frac 1 {2^{n + 1} } \sum_{k \mathop = 0}^n \paren {-1}^k \binom n k \paren {k + 1}^{-s} }$

Proof

Use Riemann Zeta Function in terms of Dirichlet Eta Function and Binomial Theorem.

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