Is the reciprocal of the Zeta function analytically continuable? As $1/\zeta(n) = \sum_{n=1}^\infty \mu(n)/{n^s}$, this does not look obvious.

Can one let me know about the functional equation of the alternating zeta function similar to the well known for the rieman function.

Consider the function $f:[0,1]\rightarrow\mathbb{R}$, where $f$ is real-analytic on the open interval $(0,1)$ $f$ is bounded on the closed interval $[0,1]$ (ie. there is some con …

I have posted a similar question in the past but let me make a final try in a simpler framework. Let $g \in C_0 ^\infty (\mathbb{R})$ be smooth and compactly supported. Define $ …

let $D_{\mathbb N}$ be the standard "n-th derivative" function is it possible to make a continuation of $D_{\mathbb N}$ to non integer values? i mean a function $D_{\mathbb R}$ s …

Suppose to start with I have a smooth function $f:\mathbb{R}^n \to \mathbb{R}$, a neighborhood $\Omega $ of $\mathbb{R}^n$ in $\mathbb{C}^n$ and that I assume $f$ to have an analyt …

Hi. I have been struggling with this question for a while now. Given a domain $\Omega \subset \mathbb{C}^n$, $\Omega^\prime = \Omega \cap \mathbb{R}^n$, and an analytic function $ …

Analytic/meromorphic continuation is a difficult problem in general. For "motivic L-functions", the idea of proving their analytic continuations by first proving their modularity g …

Fix a complex number s and a real number x, does there exist an analytic continuation of the Dirichlet series $L(s,x):=\sum_{k=1}^{\infty}\frac{\sin^2(2\pi k x)}{k^s}$ to the wh …