Editing Local zeta function (section)

==Formulation==

Given a finite field ''F'', there is, up to [[isomorphism]], only one field ''F<sub>k</sub>'' with

:<math>[ F_k : F ] = k \,</math>,

for ''k'' = 1, 2, ... .  When ''F'' is the unique field with ''q'' elements, ''F<sub>k</sub>'' is the unique field with <math>q^k</math> elements.   Given a set of polynomial equations &mdash; or an [[algebraic variety]] ''V'' &mdash; defined over ''F'', we can count the number

:<math>N_k \,</math>

of solutions in ''F<sub>k</sub>'' and create the generating function

:<math>G(t) = N_1t +N_2t^2/2 + N_3t^3/3 +\cdots \,</math>.

The correct definition for ''Z''(''t'') is to set log ''Z'' equal to ''G'',  so

:<math>Z= \exp (G(t)) \, </math>

and ''Z''(0) = 1, since ''G''(0) = 0, and ''Z''(''t'') is ''a priori'' a [[formal power series]].

The [[logarithmic derivative]]

:<math>Z'(t)/Z(t) \,</math>

equals the generating function

:<math>G'(t) = N_1 +N_2t^1 + N_3t^2 +\cdots \,</math>.