3 added 92 characters in body

Hi,

Consider the zeta function of a definable set over a finite field. More precisely, let $\varphi$ be a formula in the language of fields and let $X$ be a definable subset of $F^n$ given by $X=\{(a_1,\dots,a_n) \in F^n| F\models \varphi(a_1,\dots,a_n) \}$ where $F$ is a finite field of characteristic $p$, then $$Z_X(t)= \exp \sum_{i=1}^ \infty \frac{N_i(X)}{i} t^i,$$ where $N_i(X)= card \{ (a_1,\dots,a_n) \in F_i^n| F_i \models \varphi[a_1,\dots,a_n]\}$ with $F_i$ is the unique extension of $F$ of degree $i$ and $\varphi$ is the formula defining $X$.i$. My question is: Does the Riemann hypothesis for$Z_X(t)$hold, i.e. if we make the change of variables$t=q^{-s}$does it follow that all zeroes of$Z_X(q^{-s})$lie on the line$Re(s)=1/2$? Thank you 2 added 2 characters in body Hi, Consider the zeta function of a definable set over a finite field. More precisely, let$X$be a definable subset of$F^n$where$F$is a finite field of characteristic$p$, then $$Z_X(t)= \exp \sum_{i=1}^ \infty \frac{N_i(X)}{i} t^i,$$ where$N_i(X)= card \{ (a_1,\dots,a_n) \in F_i^n| F_i \models \varphi[a_1,\dots,a_n]}$varphi[a_1,\dots,a_n]\}$ with $F_i$ is the unique extension of $F$ of degree $i$ and $\varphi$ is the formula defining $X$.

My question is:

Does the Riemann hypothesis for $Z_X(t)$ hold, i.e. if we make the change of variables $t=q^{-s}$ does it follow that all zeroes of $Z_X(q^{-s})$ lie on the line $Re(s)=1/2$?

Thank you

1

# Riemann hypothesis for zeta function of definable sets over finite fields

Hi,

Consider the zeta function of a definable set over a finite field. More precisely, let $X$ be a definable subset of $F^n$ where $F$ is a finite field of characteristic $p$, then $$Z_X(t)= \exp \sum_{i=1}^ \infty \frac{N_i(X)}{i} t^i,$$ where $N_i(X)= card { (a_1,\dots,a_n) \in F_i^n| F_i \models \varphi[a_1,\dots,a_n]}$ with $F_i$ is the unique extension of $F$ of degree $i$ and $\varphi$ is the formula defining $X$.

My question is:

Does the Riemann hypothesis for $Z_X(t)$ hold, i.e. if we make the change of variables $t=q^{-s}$ does it follow that all zeroes of $Z_X(q^{-s})$ lie on the line $Re(s)=1/2$?

Thank you