Algebraic geometry may be regarded as research of zero set of algebraic equations system. And there are a lots of approach and result about zero set.
Riemann Hypothesis is about the non-trivial zero set of function $\zeta(x)$
They are all about zero set, but the functions belong to different type.
Any analog or parallel approach /result between zero set by algebraic equations system and zero set of Rieman zeta function?
Attempting to prove that $\zeta(s) = 0$ forms some sort of generalized variety seems like folly. (Or perhaps brilliant --- but no one has yet gotten anywhere with this thought).
However, there is a connection between algebraic varieties and generalized zeta functions. Namely, suppose $X$ is a smooth closed irreducible variety defined by polynomials $f_1, \ldots, f_d$ over a finite field $F_q$. Let $F_{q^n}$ denote the finite field with $q^n$ elements. Define $$ N_m = \# \{ u \in X(F_{q^m}) \} = \# \{ u \in F_{q^m}^d : f_i(u) = 0 \;\forall\, i \}.$$ Then we can define a "Hasse-Weil" zeta function $$ Z(X, t) = \exp \left( \sum_{m \geq 1} \frac{N_m}{m} t^m \right).$$ This function satisfies a functional equation, and it's known to satisfy its analogue of the Riemann Hypothesis. The precise form is sometimes called the Weil Conjectures (proved by Deligne).