# Signature of The Riemann Zeta Function

Let $\zeta\colon\mathbb{C}\to\mathbb{C}$ denote the Riemann zeta function, analytically defined on $\mathbb{C}$ by meromorphic continuation, where $z \in D$. Similarly, let $G$ represent the co-domain of $z$ under $\zeta(z)$, such that $\zeta(z)\in G$ if $z \in D$. Is there a necessary condition that can be imposed on $z$, such that $\zeta(z)\in\mathbb{C^+}\cup \{0\}$ (i.e. $\zeta(z)≥0$ if $z \in F\subset D$) and $\zeta(z)\in\mathbb{C^-}$ (i.e. $\zeta(z)<0$ if $z \in H\subset D$) ?

Best Regards

• of course in general near a zero of a (locally) analytic function, there are all the possible arguments, since in the neighborhood of such zero $f(\rho+\epsilon) \sim C \epsilon^n$ for some $n$ and $C \ne 0$ – reuns May 8 '16 at 4:00

• @user1952009 : "would allow multiple copies of the axes (if the root was a multiple root)" is directly addressing nonsimple zeroes. (While it still referenced "$\Bbb{R}^+$" and "$\Bbb{R}^-$" and before it referenced "$\Bbb{C}^+$" and "$\Bbb{C}^-$", i.e. when it was still sensical :) The questioner wants to identify the set of points where zeta is nonnegative real value or nonpositive real valued. – Eric Towers May 8 '16 at 4:04