Given F, there is, up to isomorphism, just one field Fk with [Fk:F] = k, for k = 1,2, ... . Given polynomial equations - or an algebraic variety V - defined over F, we can count the number Nk of solutions in Fk; and create the generating function
G(t) = N1.t + N2.t2/2 + ... .
The correct definition for Z(t) is to make log Z equal to G, and so Z = exp(G); we will have Z(0) = 1 since G(0) = 0, and Z(t) is a priori a formal power series.
For example, assume all the Nk are 1 (this happens for example if we start with an equation like X = 0, so that geometrically we are taking V a point). Then
G(t) = log(1 - t)
is the expansion of a logarithm (for |t| < 1). In this case we have
Z(t) = 1/(1 - t).
To take something more interesting, let V be the projective line over F. If F has q elements, then this has q + 1 points, including as we must the one point at infinity. Therefore we shall have
Nk = qk + 1
and
G(t) = log(1 - t) + log(1 - qt),
for |t| small enough.
In this case we have
Z(t) = 1/{(1 - t)(1 - qt)}.
The relationship between the definitions of G and Z can be explained in a number of ways. In practice it makes Z a rational function of t, something that is interesting even in the case of V an elliptic curve over finite field.
It is the functions Z that are designed to multiply, to get global zeta functions. Those involve different finite fields (for example the whole family of fields Z/p.Z as p runs over all prime numbers. In that relationship, the variable t undergoes substitution by p-s, where s is the complex variable traditionally used in Dirichlet series. This explains too why the logarithmic derivative with respect to s is used.
With that understanding, the products of the Z in the two cases come out as ζ(s) and ζ(s)ζ(s-1).
For projective curves over F that are non-singular, it can be shown that
Z(t) = P(t)/{(1 - t)(1 - qt)},
with P(t) a polynomial, of degree 2g where g is the genus of C. The Riemann hypothesis for curves over finite fields states that the roots of P have absolute value q-1/2, where q = |F|.
For example, for the elliptic curve case there are two roots, and it is east to show their product is q-1. Hasse's theorem is that they have the same absolute value; and this has immediate consequences for the number of points.
Weil proved this for the general case, around 1940 (Comptes Rendus note, April 1940): he spent much time in the years after that, writing up the algebraic geometry involved). This led him to the general Weil conjectures, finally proved a generation later. See etale cohomology for the basic formulae of the general theory.Riemann hypothesis for curves over finite fields