# Category Archives: Dirichlet eta function

## Relation between Zeta and Dirichlet eta functions proof

$$\eta(s) = \left( 1-2^{1-s} \right) \zeta(s)$$ $$\textit{proof}$$ We will start by the RHS $$\left( 1-2^{1-s} \right) \zeta(s) = \zeta(s) – 2^{1-s} \zeta(s)$$ Which can be written as sums of series $$\sum_{n=1}^\infty \frac{1}{n^s} – \frac{1}{2^{s-1}}\sum_{n=1}^\infty \frac{1}{n^s}$$ \sum_{n=1}^\infty \frac{1}{n^s} – 2\sum_{n=1}^\infty … Continue reading