Let $X$ be an extremally disconnected (i.e. such that the closure of open sets is open) compact Hausdorff space. Then
$*_1$ $C(X)$ is the space of continuous functions $f: X \to \mathbb{C}$,
$*_2$ $C^+(X)$ is the space of continuous functions $f: X \to \mathbb{S}^2 = \mathbb{C} \cup \{\infty\}$ such that the pre-image of $\infty$ is nowhere dense ($\mathbb{S}^2$ is the one point compactification of $\mathbb{C}$).
Suppose $\mathbb{B}$ is a complete Boolean algebra, and consider the Boolean valued universe $V^{\mathbb{B}}$. Let $St(\mathbb{B})$ be the Stone space of $\mathbb{B}$. Then there is a correspondence between:
$\star_1$ the family of $\mathbb{B}$-names for complex numbers in the boolean valued model $V^{\mathbb{B}}$: $\dot{\mathbb{B}} = \{\tau \in V^{\mathbb{B}}: \parallel \tau \text{~is a complex number~}\parallel_{\mathbb{B}} = 1_{\mathbb{B}} \}$
and
$\star_2$ $C^+(St(\mathbb{B}))$.
Remark. Let $\mathbb{B}$ be the complete boolean algebra given by Lebesgue measurable sets modulo Lebesgue null sets. Then $C(St(\mathbb{B}))$ is isomorphic to $L^\infty(\mathbb{R})$.
For more details, see [1], [2] below.
These results, motivate us to ask the following question:
Question. Suppose $\mathbb{B}$ is a complete Boolean algebra, or even a simple one, say the complete boolean algebra given by Lebesgue measurable sets modulo Lebesgue null sets. Is there a statement $RH(\mathbb{B})$ such that:
$V^{\mathbb{B}}\models $the Riemann hypothesis holds $\iff~ RH(\mathbb{B})$ holds for $C^+(St(\mathbb{B}))$.
Remark As the Riemann hypothesis is a $\Pi_1$ statement, by the Shoenfield absoluteness theorem, its truth does not change by forcing. Thus for any complete Boolean algebra $\mathbb{B}$, we have $V^{\mathbb{B}}\models $``the Riemann hypothesis holds'' iff the Riemann hypothesis holds.
[1] Vaccaro, Andrea; Viale, Matteo; Generic absoluteness and boolean names for elements of a Polish space, Boll. Unione Mat. Ital. 10 (2017), no. 3, 293–319.
[2] Viale, Matteo; Forcing the truth of a weak form of Schanuel's conjecture, Confluentes Mathematici. 8 (2016), no. 2, 59–83.
