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I'm currently reading through Ireland and Rosen's "A Classical Introduction to Modern Number Theory", and I feel I'm missing the point of Chapter 11 on (local) zeta functions. When I see that they define the zeta function associated with a homogenous polynomial $f$ in $n+1$ variables in terms of power series, and then prove several facts about this object, it all feels fairly unmotivated. Before delving any further, my question is: What is the simplest application of the local zeta function? i.e. what it the simplest or easiest number theoretic fact/result that can be proved using the local zeta function that can't be better proved by other means?

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A bit more background: If $f(y) \in F[y_0,\ldots,y_n]$ is a homogenous polynomial with coefficients in a finite field $F$ with $q$ elements, let $N_s$ be the number of zeroes of $f$ in the projective space $P^n(F_s)$, where $F_s$ is the unique (up to isomorphism) extension of $F$ with $q^s$ elements. Now let $$Z_f(u) := \exp \left( \sum_{s=1}^\infty N_s u^s/s \right).$$

Throughout the chapter various computations are done, computing $Z_f(u)$ directly for some basic choices of $f$, but each of these calculations goes via computing $N_s$ directly, and then in the end packaging $(N_s)_{s \geq 1}$ into the power series $Z_f(u)$. And I'm left wondering what the point is! I should say, in spite of the slightly provocative title of my question, I don't doubt that there is value in studying the expression $Z_f(u)$, only that it's value is as of yet unclear to me.

I might also mention that I've worked through the beautiful proof of the prime number theorem using the Riemann zeta function, but it was my impression that the utility here relies on the Euler product formula. Is there an analogue for local zeta functions wherein contour integrals yield information about the asymptotics of $N_s$ for large $s$? I'd be more than satisfied with a local zeta function parallel to the Riemannzeta-PNT correspondence as evidence of the utility of the former. And I would be very grateful if someone could point me to a reference with which I could work through the details of such a calculation (preferably without too much Zariski topology or Tate's thesis to scare me off!).

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    $\begingroup$ One application is the solution of the Riemann hypothesis for curves over finite fields, see wikipedia. $\endgroup$
    – Dietrich Burde
    Commented May 24 at 9:26
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    $\begingroup$ Several classical problems in analytic number theory have their counterparts in the function field side (the ring of polynomials over a finite field). Particularly in the case of univariate polynomials: the Riemann hypotheses, Dirichlet's theorem of the distribution of primes into residue classes, Artin's conjecture (a conjecture in $\Bbb{Z}$, a theorem in $\Bbb{F}_q[x]$, PNT (a theorem for both). IIRC the twin prime conjecture is open for both. Take a look at Michael Rosen's book Number theory in function fields. $\endgroup$
    – Jyrki Lahtonen
    Commented May 24 at 12:50

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