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 $i$.


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



Riemann hypothesis for zeta function of definable sets over finite fields
No.


The zeta function of the set $\{x|x \neq 0\}$ over $\mathbb F\_p$ is:


$e^{\sum\_{i=1}^\infty \frac{p^i-1}{i} t^i} = \frac{1-t}{1-pt}=\frac{1-p^{-s}}{1-p^{1-s}}$


It has a zero where $s=0$. 


More generally, every affine algebraic curve minus a point has a zeta function that is zero at $s=0$.


