Let $\{ \lambda_i | \ i \in \mathbb{N}_0\} \subset \mathbb{R}^+$ the spectrum of a Laplacian on a compact closed Riemannian manifold.
I have to show that the function
$ f(s) := \frac{1}{\Gamma(s)} \int_0^1 t^{s-1} ( \sum_{i=1}^\infty e^{-\lambda_1t}) \mathrm{dt}$
is holomorphic. In a book I found the following statement:
Since the Gamma-function is never zero and f is of the form $ \frac{1}{\Gamma(s)} \int_0^1 t^{s-1} O(e^{-\lambda_1t}) \mathrm{dt}$, where $\lambda_ 1$ refers to the first nonzero Eigenvalue, f is holomorphic"
Can someone explain exactly how this argumentation proves the holomorphy?