For $Re(s) > 0$, the Dirichlet eta function has the series representation $\eta(s) = \sum_{n=1}^\infty (-1)^{n+1} n^{-s}$. It extends analytically to the whole complex plane, and we have the following representation : $$\boxed{\forall s \in \mathbb{C}, \qquad -\eta(s) = \lim_{z \ \to \ -1, |z| < 1} \sum_{n=1}^\infty z^{n} n^{-s}}$$
Question : do you know other proofs, preferably more general ?
- Let $\displaystyle f(s,z ) = \sum_{n=1}^\infty z^{n} n^{-s} $ (the Polylogarithm). Note it is analytic in $s$ and $z$ for $|z| < 1$. On $|z| = 1$, define $f(s,z) = \lim_{r \to 1^-}f(s,rz)$, we will prove it exists for $z \ne 1$ and that it stays analytic in $s$.
Summing by parts, for $Re(s) > 1, |z| \le 1$ : $$f(s,z) = \sum_{n=1}^\infty \frac{z^{n+1}-z}{z-1} (n^{-s}-(n+1)^{-s})= \frac{-z}{z-1} + \frac{z}{z-1}\sum_{n=1}^\infty z^n (n^{-s}-(n+1)^{-s})$$ using $n^{-s}-(n+1)^{-s} = s \int_n^{n+1} x^{-s-1}dx \sim s n^{-s-1}$ as $n \to \infty$, we get that $z\sum_{n=1}^\infty z^n (n^{-s}-(n+1)^{-s})$ is a representation of $(z-1)(f(s,z)+z)$ valid for $Re(s) > 0, |z| \le 1$.
Then apply the same argument inductively : letting $a_n^0(s) = n^{-s}$, $a_n^{k+1}(s) = a_n^k(s)-a_{n+1}^k(s)$, $\ a_n^k(s) \sim s^k n^{-s-k}$ as $n \to \infty$. Define $$h_0(s,z) = f(s,z),\qquad h_{k+1}(s,z) = (z-1)(h_k(s,z)+z \, a_1^k(s))$$ so that $z^{k+1}\sum_{n=1}^\infty z^{n} (a_n^k(s)- a_{n+1}^k(s))$ is a representation of $h_{k+1}(s,z)$ valid for $Re(s) > -k, |z| \le 1$.
- Note that for all $m \ge 0$ : $$\frac{d^m}{d s^m} a_n^k(s) \sim \frac{d^m}{d s^m} s^k n^{-s-k} \qquad \text{as }n \to \infty$$ so that $ z^k\sum_{n=1}^\infty z^{n}\frac{d^m}{d s^m} a_n^k(s)$ is also a valid representation of $\frac{d^m}{d s^m} h_{k}(s,z)$, proving the analyticity for $Re(s) > 1-k, |z| \le 1$.
- Hence, defining $\tilde{h}_0(s) = -\eta(s),\ \tilde{h}_{k+1}(s) = 2 (\tilde{h}_{k}(s)- a_1^k(s))$ we get, for $Re(s) > 1-k$ : $$h_{k}(s,-1) = \tilde{h}_k(s)$$
- And since $h_k(s,z)$ is clearly continuous in $z$ for $|z| \le 1, Re(s) > 1-k$ : $\tilde{h}_k(s) = \lim_{z \,\to\, -1}h_{k}(s,z)$. Being true for every $k$, it means that $$\forall s \in \mathbb{C}, \qquad\lim_{z \,\to\, -1} f(s,z) = -\eta(s)$$
Also, it should prove $f(s,e^{2i \pi x})$ is entire in $s$ whenever $x \in \mathbb{R}\setminus\mathbb{Z}$. Any ideas or references are welcome : extending $f(s,z)$ analytically beyond $|z| \le 1$, a functional equation for $f(s,z)$..
