$\zeta$ Function

The $\zeta$ Function

We start with the following definition for the Zeta function:

$$ζ(n) := \sum\limits_{k=1}^{\infty} \frac{1}{k^n}, \quad n > 1$$

More Special Sums

Here are some more sums that are special enough to have names.

$\begin{eqnarray} \lambda(n) &=& \sum\limits_{k=1}^\infty \frac{1}{(2k-1)^n} &=& 1 + \frac{1}{3^n} + \frac{1}{5^n} + \frac{1}{7^n} + \frac{1}{9^n} + \cdots &=& \left(1 – \frac{1}{2^n} \right) \zeta(n) \\ \eta(n) &=& \sum\limits_{k=1}^\infty \frac{(-1)^{k+1}}{k^n} &=& 1 – \frac{1}{2^n} + \frac{1}{3^n} – \frac{1}{4^n} + \frac{1}{5^n} – \cdots &=& \left(1 – \frac{1}{2^{n-1}} \right) \zeta(n) \\ \beta(n) &=& \sum\limits_{k=0}^\infty \frac{(-1)^k}{(2k+1)^n} &=& 1 – \frac{1}{3^n} + \frac{1}{5^n} – \frac{1}{7^n} + \frac{1}{9^n} – \cdots &=& 4^{-n} \left(\zeta\left(n,\frac{1}{4}\right) – \zeta\left(n,\frac{3}{4}\right) \right)\\ \end{eqnarray}$