When I read it for the first time, I found the whole slog towards proving the Prime Number Theorem and the final success to be magnificent. So I am curious about more general results.
We talk of complex $\zeta$-functions and $L$-functions. As a preliminary list, we fix the following list. But feel free to add to it.
$1$. Riemann $\zeta$-function.
$2$. Dedekind $\zeta$-function for a number field.
$3$. Artin $L$-functions for a character of the Galois group of some number field.
$4$. Zeta functions of algebraic varieties over number fields; for getting analytic continuation up to $s = 0$, we for instance fix the zeta function of an elliptic curve defined over $\mathbb{Q}$ which is modular in the sense of Eichler-Shiumura.
I am worried about $4$ here, since the requirement that the zeta function has a pole at $s=1$ is not fulfilled, and thus we fail to capture the main term from the residue there. Is something possible in this case? If so, we can consider that and also more general zeta functions of algebraic varieties over any finitely generated field, zeta functions of Galois representations, etc.. which have analytic continuation up to $s=0$ and the question could be extended to those cases as well.
We have the original prime number theorem, in the following form,
$\pi (x) = \frac{x}{log x} + O(error\ term)$.
which is proved by integration of the Riemann zeta function along a rectangular contour including $s = 1$, and letting the vertical edges get longer and longer, and estimating the integrals. The reference I have in mind is Ram Murty, Problems in Analytic Number Theory, or, J. Ayoub's book.
Now the questions:
What is the known about a similar prime number theorem for more general zeta and $L$-functions?
I could imagine that it will be pretty straightforward for the Dedekind zeta function. But then I am curious about the error term.
For the rest of the cases, does a similar proof of the PNT carry over? More importantly,
What is the "meaning" of the prime number theorem in these general cases?
Finally,
What are some applications(to other problems) of such a prime number theorem proved with a good error term?
The applications of the original PNT are of course well-known, as for instance given in the books of Titchmarsch and Heath-Brown, or Ivic, or in the more modern book of Iwaniec and Kowalski.
Best Answer
Hi Anweshi,
Since Emerton answered your third grey-boxed question very nicely, let me try at the first two. Suppose $L(s,f)$ is one of the L-functions that you listed (including the first two, which we might as well call L-functions too). (For simplicity we always normalize so the functional equation is induced by $s\to 1-s$.) This guy has an expansion $L(s,f)=\sum_{n}a_f(n)n^{-s}$ as a Dirichlet series, and the most general prime number theorem reads
$\sum_{p\leq X}a_f(p)=r_f \mathrm{Li}(x)+O(x \exp(-(\log{x})^{\frac{1}{2}-\varepsilon})$.
Here $\mathrm{Li}(x)$ is the logarithmic integral, $r_f$ is the order of the pole of $L(s,f)$ at the point $s=1$, and the implied constant depends on $f$ and $\varepsilon$.
Let's unwind this for your examples.
1) The Riemann zeta function has a simple pole at $s=1$ and $a_f(p)=1$ for all $p$, so this is the classical prime number theorem.
2) The Dedekind zeta function (say of a degree d extension $K/\mathbb{Q}$) is a little different. It also has a simple pole at $s=1$, but the coefficients are determined by the rule: $a(p)=d$ if $p$ splits completely in $\mathcal{O}_K$, and $a(p)=0$ otherwise. Hence the prime number theorem in this case reads
$|p\leq X \; \mathrm{with}\;p\;\mathrm{totally\;split\;in}\;\mathcal{O}_K|=d^{-1}\mathrm{Li}(x)+O(x \exp(-(\log{x})^{\frac{1}{2}-\varepsilon})$.
This already has very interesting applications: the fact that the proportion of primes splitting totally is $1/d$ was very important in the first proofs of the main general results of class field theory.
3) If $\rho:\mathcal{G}_{\mathbb{Q}}\to \mathrm{GL}_n(\mathbb{C})$ is an Artin representation then $a(p)=\mathrm{tr}\rho(\mathrm{Fr}_p)$. If $\rho$ does not contain the trivial representation, then $L(s,\rho)$ has no pole in neighborhood of the line $\mathrm{Re}(s)\geq 1$, so we get
$\sum_{p\leq X}\mathrm{tr}\rho(\mathrm{Fr}_p)=O(x \exp(-(\log{x})^{\frac{1}{2}-\varepsilon})$.
The absence of a pole is not a problem: it just means there's no main term! In this particular case, you could interpret the above equation as saying that "$\mathrm{tr}\rho(\mathrm{Fr}_p)$ has mean value zero.
4) For an elliptic curve, the same phenomenon occurs. Here again there is no pole, and $a(p)=\frac{p+1-|E(\mathbb{F}_p)|}{\sqrt{p}}$. By a theorem of Hasse these numbers satisfy $|a(p)|\leq 2$, so you could think of them as the (scaled) deviation of $|E(\mathbb{F}_p)|$ from its "expected value" of $p+1$. In this case the prime number theorem reads
$\sum_{p\leq X}a(p)=O(x \exp(-(\log{x})^{\frac{1}{2}-\varepsilon})$
so you could say that "the average deviation of $|E(\mathbb{F}_p)|$ from $p+1$ is zero."
Now, how do you prove generalizations of the prime number theorem? There are two main steps in this, one of which is easily lifted from the case of the Riemann zeta function.
Prove that the prime number theorem for $L(s,f)$ is a consequence of the nonvanishing of $L(s,f)$ in a region of the form $s=\sigma+it,\;\sigma \geq 1-\psi(t)$ with $\psi(t)$ positive and tending to zero as $t\to \infty$. So this is some region which is a very slight widening of $\mathrm{Re}(s)>1$. The proof of this step is essentially contour integration and goes exactly as in the case of the $\zeta$-function.
Actually produce a zero-free region of the type I just described. The key to this is the existence of an auxiliary L-function (or product thereof) which has positive coefficients in its Dirichlet series. In the case of the Riemann zeta function, Hadamard worked with the auxiliary function $ A(s)=\zeta(s)^3\zeta(s+it)^2 \zeta(s-it)^2 \zeta(s+2it) \zeta(s-2it)$. Note the pole of order $3$ at $s=1$; on the other hand, if $\zeta(\sigma+it)$ vanished then $A(s)$ would vanish at $s=\sigma$ to order $4$. The inequality $3<4$ of order-of-polarity/nearby-order-of-vanishing leads via some analysis to the absence of any zero in the range $s=\sigma+it,\;\sigma \geq 1-\frac{c}{\log(|t|+3)}.$ In the general case the construction of the relevant auxiliary functions is more complicated. For the case of an Artin representation, for example, you can take $B(s)=\zeta(s)^3 L(s+it,\rho)^2 L(s-it,\widetilde{\rho})^2 L(s,\rho \otimes \widetilde {\rho})^2 L(s+2it,\rho \times \rho) L(s-2it,\widetilde{\rho} \times \widetilde{\rho})$. The general key is the Rankin-Selberg L-functions, or more complicated L-functions whose analytic properties can be controlled by known instances of Langlands functoriality.
If you'd like to see everything I just said carried out elegantly and in crystalline detail, I can do no better than to recommend Chapter 5 of Iwaniec and Kowalski's book "Analytic Number Theory."