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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.

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3 Answers

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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.

  1. 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.

  2. 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."

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Ahh! It's all in Iwaniec and Kowalski! Which I myself cited! This should teach me to look into my own references before I jump and ask questions. –  Anweshi Feb 5 '10 at 22:50
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In case 2 it should be $\#Gal(K^{gc}/\mathbb{Q})^{-1} Li(x)$, or assume that $K$ is already galois. –  Dror Speiser Jul 7 '10 at 7:29
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If you want bounds for the smallest prime with a certain behaviour in a given extension field, you'll need Cebotarev with error term estimates. This is important even in classical number theory, e.g. estimating the smallest quadratic non-residue mod $p$, in terms of $p$, for instance. Likewise, if you want to know the smallest $p$ for which the trace of Frobenius $a_p$ of a fixed elliptic curve $E$ is in an given interval $[\alpha\sqrt{p},\beta\sqrt{p}]$, you want PNT for the L-function of $E$ and so on.

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The one-line answer to your question is the Sato--Tate conjecture. You might google this to find expository discussion, and looks in Serre's Abelian l-adic reps. book to find a proof in the context of elliptic curves modulo properties of motivic L-functions (now established -- by Clozel, Harris, Shepherd-Barron, Shin, and Taylor). There is an analog of Sato--Tate for any irreducible motive over a number field; precise statements are probably in one of the articles in the motives volumes.

For this, and for other applications too, it is important that irreducible non-trivial motives (conjecturally) have $L$-functions with no pole; only the trivial motive contributes poles.

For applications of the PNT for Dedekind zeta function, you might look at Hooley's proof of the Artin primitive root conjecture, conditional on GRH. (He needs GRH because he adds up an infinity of error terms, and needs a good enough estimate on them that he can still control their sum.)

Added: Also, for Artin $L$-functions, the PNT is Cebotarev density (although this was actually proved earlier, and if I understand the history correctly, was a source of insipiration for Artin).

For Dirichlet $L$-functions, it is Dirichlet's theorem on primes in arithmetic progression.

Finally, let me remark that the condition that only the trivial motive (and its Tate twists) should give an $L$-functions with a pole (i.e. the only pole in $L$-functions should come from factors of the form $\zeta(s)$, or $\zeta(s + n)$) is a Fourier-theoretic condition, analogous to the theorem that for an irrep. of a group, the average value of the character is zero except for the trivial representation. (To understand what this means, examine Dirichlet's proof of this theorem, and see how the character-theoretic fact I just mentioned interacts with the study of the poles and zeroes of the $L$-functions.) (To be slightly more precise, one should combine this condition on poles with a corresonding condition on zeroes, namely that, after normalizing so that functional equation relates $s$ and $1 - s$, so that the critical line is $Re s = 1/2$, it should be the case that there are no zeroes on the line $Re s = 1$.)

Added in response to Anweshi's comments: I should say that when I wrote Dirichlet's theorem and Cebotarev, I mean the quantative versions, giving correct asymptotics.

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Deaer Emerton. Dirichlet's theorem on arithmetic progression follows just from the fact that the Dedekind zeta function has a pole at $s = 1$. It is not an analogue of PNT; one does not have to do any of contour integration I described, or difficult and hard estimates. It is a much simpler theorem in analytic number theory. Chebotarev is just the generalization of Dirichlet's theorem, not of PNT. The Sato-Tate is looking more closely at the error term in the Weil bound, how is this a generalization of PNT?? –  Anweshi Feb 5 '10 at 18:05
    
Dear prof. Emerton., Your answer was very helpful nevertheless, and I have upvoted it. –  Anweshi Feb 5 '10 at 18:08
    
I mean the versions of Dirichlet and Cebotarev with the correct asymptotics. As for Sato--Tate, an examination of the discussion in Serre's book will answer your question. –  Emerton Feb 5 '10 at 18:42
    
To add another detail to Matt's clarification, even the Riemann zeta-function tells you something from its pole at s = 1: there are infinitely many primes. If you want prime asymptotics (in the sense of natural density) then you have to deal with behavior on the whole line Re(s) = 1, and likewise for Dirichlet's theorem in its qualitative (or analytic density) form vs. in its natural density form. –  KConrad Feb 6 '10 at 2:22
    
Thanks Keith! That puts it very nicely. –  Emerton Feb 6 '10 at 2:35
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