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(I'll keep this one short)

Given a Dirichlet series

$$g(s)=\sum_{k=1}^\infty\frac{c_k}{k^s}$$

where $c_k\in\mathbb R$ and $c_k \neq 0$ (i.e., the coefficients are a sequence of arbitrary nonzero real numbers), and assuming that $g(s)$ can be analytically continued, does it follow that $g(s)$ possesses a critical strip containing its nontrivial zeroes?

If this does not generally hold, what restrictions should there be on the $c_k$ for $g(s)$ to possess a critical strip?

(My attempts at searching bring too much stuff on Riemann $\zeta$, with only a quick mention of general Dirichlet series; pointers to the literature would be appreciated.)

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This other answer of mine constructs a Dirichlet series with infinitely many poles, but no "critical strip". math.stackexchange.com/questions/3405/…. To get a critical strip, you really need a functional equation which reflects the series (or a Meromorphic extension) about a critical line. –  George Lowther Dec 23 '10 at 17:42
    
@George: so only the "assuming that $g(s)$ can be analytically continued" (adding "to the left of the complex plane") guarantees the existence of a strip, then... –  J. M. Dec 24 '10 at 0:13
    
Not sure I follow you. Didn't my comment suggest the opposite - that an analytic (or meromorphic) extension is not enough? And what does "to the left of the complex plane" mean? –  George Lowther Dec 24 '10 at 0:37
    
@George: Yes, sorry, I was being dense there. So it's "no reflection relation, no critical strip" then. I guess the question should now be on what's needed for a Dirichlet series to have a reflection relation... –  J. M. Dec 24 '10 at 0:55
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3 Answers

up vote 26 down vote accepted

In the comments section to Willie Wong's answer, the following Dirichlet series came up: the Riemann $\zeta$-function, Dirichlet $L$-functions, and Ramanujan's series $\sum_{n \geq 1}\tau(n) n^{-s}$, where $\tau(n)$ is the coefficient of $q^n$ in $\Delta(q) = q\prod_{n=1}^{\infty} (1-q^n)^{24}$.

First note that the $\zeta$-function is a special case of a Dirichlet $L$-function (it is the $L$-function of the trivial character).

Now what is it that Dirichlet $L$-functions and Ramanujan's series have in common? Well, they are all automorphic $L$-functions.

An automorphic form (for the group $\mathrm{GL}_n$ over $\mathbb Q$; there are generalizations where $\mathbb Q$ is replaced by an arbitrary number field $F$ and $\mathrm{GL}_n$ is replaced by an arbitrary reductive group, but to simplify the explanations, I will focus just on the simplest level of generality here) is a function on the product $\mathrm{GL}_n(\mathbb R)\times \mathrm{GL}_n(\mathbb Z/N\mathbb Z)$ for some integer $N \geq 1$ which is

  • invariant under the natural (diagonal) action of $\mathrm{GL}_n(\mathbb Z)$;

  • grows moderately at infinity with respect to the $\mathrm{GL}_n(\mathbb R)$-coordinates;

  • satisfies a suitable differential equation in the $\mathrm{GL}_n(\mathbb R)$-coordinates.

Rather than explaining the generalities in more detail (they can be found in many places), I think it's better to illustrate them:

E.g. Dirichlet characters arise in the case $n = 1$: they are defined as functions on $(\mathbb Z/N\mathbb Z)^{\times} =: \mathrm{GL}_1(\mathbb Z/N\mathbb Z)$, and so we can make them into functions on $\mathrm{GL}_1(\mathbb R)\times \mathrm{GL}_1(\mathbb Z/N\mathbb Z)$ by defining them to be trivial on the $\mathbb R^{\times}$-coordinate.

E.g. If $f(\tau)$ is a holomorphic modular form of weight $k$ and level one (where $\tau$ is an upper half-plane variable as usual), we can make $f$ into a function on $\mathrm{GL}_2(\mathbb R)$ by first identifying this matrix group with the collection of bases of $\mathbb R^2$, then identifying $\mathbb R^2$ with $\mathbb C$, and then defining, for any $\mathbb R$-basis $\omega_1,\omega_2$ of $\mathbb C$, $f(\omega_1,\omega_2) := \omega_2^{-k} f(\omega_1/\omega_2)$. (This presumes that $\omega_1/\omega_2$ is in the upper half-plane rather than the lower, for simplicity.) Thus we get a function of the required kind (with $N = 1$).

The usual modularity condition becomes invariance under $\mathbb GL_2(\mathbb Z)$. The moderate growth condition becomes the condition that the Fourier expansion of $f$ involves only non-negative powers of $e^{2 \pi i \tau}$. The differential equation is the Cauchy--Riemann equation expressing holomorphy of $f$.

Higher level modular forms will involve values of $N$ that are $> 1$.

E.g. Maass forms are similar to the preceding example, except that now the differential equation expresses that a Maass form is an eigenvector of the Laplacian.

For any fixed $n$ and fixed $N$, we have Hecke operators acting on the space of automorphic forms, labelled by primes $p$ not dividing $N$, and so we can consider Hecke eigenforms. In the case of Dirichlet characters, the fact that they are characters of $(\mathbb Z/N\mathbb Z)^{\times}$ (rather than just arbitrary functions) can be reinterpreted as saying that they are Hecke eigenforms.

Of course, Ramanujan's $\Delta$ is well-known to be a Hecke eigenform of weight $12$ and level $1$.

Given an automorphic Hecke eigenform we can use the Hecke eigenvalues to make an Euler product Dirichlet series, which will give Dirichlet $L$-functions for Dirichlet characters, and Ramanujan's Dirichlet series for $\Delta$. (In the Dirichlet character case, if a prime $p$ divides the conductor $N$, we just have a trivial factor in the Euler product for that prime; when $n > 1$, and $N > 1$, it is a bit more of a battle to figure out what Euler factors to put in at the primes dividing $p$, but it can be done.)

Actually, it is better to restrict to cuspidal automorphic Hecke eigenforms. Cuspidal is a vacuous condition when $n = 1$ (i.e. in that case we agree to call everything cuspidal), and when $n > 1$ we replace "moderate growth at infinity" by "rapid decay at infinity", suitably interpreted. I'll assume that all my eigenforms are cuspidal form now on. (E.g. $\Delta$ is cuspidal.)

In this way we get a natural class of $L$-functions which have:

  • meromorphic continuation to the whole complex plane, which is in fact holomorphic with the sole exception of Riemann's $\zeta$.

  • Functional equation with completely understood $\Gamma$-factors. E.g. for a weight $k$ modular form of level one, if the $p$th Hecke eigenvalue is $a_p$, then the $L$-function is $\prod_p (1 - a_p p^{-s} + p^{k - 1 - 2s})^{-1},$ and the functional equation relates $s$ and $k - s$. For $\Delta,$ I've already noted that $k = 12$. (In general the functional equation relates the $L$-series of an automorphic eigenform with $L$-series of its "complex conjugate" suitably understood, just as in the case of Dirichlet characters that are not necessarily real valued.)

  • Conjecturally, they should all satisfy the analogue of RH, i.e. all non-trivial zeroes should lie on the critical line, in the centre of the critical strip.

Note incidentally that it is easy to change the apparent form of the functional equation. E.g. if we make a change of variable $s \mapsto s + 11/2$ in Ramanujan's series, then the functional equation will become $s \mapsto 1 -s$ rather than $s \mapsto 12 -s $, and the critical line will be $\Re s = 1/2$, just as in the $\zeta$-function case.

All cuspidal automorphic $L$-functions can be renormalized in a similar way, so that the symmetry of the functional equation is $s \mapsto 1 -s$. This is called unitary normalization, and is common in the automorphic forms literature.

Up to rescaling, there are only countably many automorphic eigenforms altogether (just because if we fix the level $N$ and (appropriately generalized version of) the weight the space of automorphic forms is finite dimensional) and so altogether we are talking about a very special class of just countably many Dirichlet series, but these seem to be the ones that naturally generalize $\zeta(s)$.

By the way, this general point of view is due to Langlands, and forms a part of the general Langlands program.

Another point of view was given by Selberg, which focuses more on capturing the analytic properties necessary for getting good properties of a Dirichlet series, rather than beginning from a conceptual construction (as in the automorphic point of view). Namely, he introduced the Selberg class of Dirichlet series. Note that part of his axioms are an Euler product, analytic continuation, and functional equation.

My sense is, though, that people expect the Selberg class of Dirichlet series to more-or-less coincide with the class of automorphic $L$-functions, so I think it is just two points of view on the same question: Langlands is showing how to construct "good" Dirichlet series, and Selberg is writing down the properties a "good" Dirichlet series should satisfy. It turns out that "good" Dirichlet series are so special, though, that however you try to pick them out, you seem to end up with the same very special collection, namely the automorphic ones.

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Thanks for this nice answer Matt! A follow-up question: are there Dirichlet series that cannot be classified as (cuspidal) automorphic L -functions, yet still possess a critical line of nontrivial zeroes? –  J. M. Dec 24 '10 at 9:51
    
@J.M.: Dear J.M., Not that I know of. My suggestion is that if no-one else here can answer that question (perhaps after posing it as a separate question, so that everyone who might answer it sees it), that you pose it on MO, where it might have a better chance of being seen by the right expert. –  Matt E Dec 24 '10 at 13:51
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@MattE: Very nice answer, I enjoyed reading it. Do you know of any good expository papers on the Langlands program or automorphic forms? I am interested interested in learning more. Also, since I am taking a modular forms course next term, it helps to have these broad ideas in mind of what researchers interested in to motivate learning the basics of the subject. –  Eric Naslund Dec 31 '11 at 19:21
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@Eric: Dear Eric, I should add that of course there are many papers trying to expose various aspects of the Langlands program, but I'm not sure my recommendation would be any better than doing a google search. I'm always at a bit of a loss as to what to recommend to my students to get started, because of the multi-faceted nature of the program. You could actually start with Langlands's own article Problems in the theory of automrphic forms, where he lays out his basic ideas about functoriality. (All his papers are available on the IAS website --- just google.) Best wishes, –  Matt E Jan 1 '12 at 1:24
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@Eric: Dear Eric, For the big picture point of view, I think the most important thing is to get some idea of both functoriality (relating automorphic forms on different groups) and reciprocity (relating automorphic forms and motives). Both involve understanding the role of the dual group, and they are different, but closely related. My posts may give some indications of what's going on, and Langlands has various expository articles (look at his collected works). You can also look in the Corvalis volumes --- some articles are more accessible than others. ... –  Matt E Jan 1 '12 at 1:28
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$g$ will always have a half-plane free from zeroes (cf. e.g. Titchmarsh, Theory of functions, Section 9.6). This means a functional equation would be sufficient to guarantee a critical strip (essentially calling the other zeroes trivial by definition).

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Possible keywords include Dirichlet L-functions or the generalized Riemann hypothesis. For the case where $c_k$ are the values of a Dirichlet character, in certain situations the existence of a critical strip is known. See the Wikipedia article for references.

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@J.M.: Dear J.M., By the Ramanujgan Dirichlet series do you mean $\sum_n \tau(n) n^{-s}$? I would be happy to answer questions about this, if this is what you have in mind. Also, I would also call this an $L$-function (it is a special case of an automorphic $L$-function). Let me know if this is a point you would like clarified, and I'd be happy to (try to!) do so. Cheers, –  Matt E Dec 23 '10 at 19:48
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J.M. and Willie: the Ramanujan Dirichlet series has a functional equation relating s and 12-s, so the critical line is Re(s) = 6. For the Riemann zeta-function and Dirichlet L-functions, the functional equation relates s and 1-s, with critical line Re(s) = 1/2. Everything is consistent: if the functional equation relates s and k-s for some constant k, the critical line should be Re(s) = k/2. –  KCd Dec 23 '10 at 23:25
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@Matt and KCd: Yes, I had the Dirichlet series whose coefficients are $\tau(n)$ in mind. I was using it as an example... so, what would be the form of the hypothesis that covers $\zeta$, the Ramanujan series, and Dirichlet L-series as special cases? Or probably this is the question that should have been asked: what restrictions on the coefficients of the Dirichlet series should be applied so that one has a functional equation (thus determining k in KCd's comment, and the existence of a critical strip)? –  J. M. Dec 24 '10 at 0:23
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Dear J.M, "Automorphic $L$-function" captures the Riemann $\zeta$-function, Dirichlet $L$-functions, Ramanujan's Dirichlet series, the $L$-functions attached to Hecke characters (aka Grossencharacters), and many more things. They are characterized not so much by an explicit restriction on the coefficients, but by the fact that they are all constructed in a certain manner from automorphic forms. It is their automorphic nature that is giving them their functional equation, critical strip, etc. A highly related concept is that of the Selberg class of Dirichlet series (which has a ... –  Matt E Dec 24 '10 at 3:30
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... wikipedia entry). If this seems like the sort of answer you're looking for, let me know, and I can write it up more carefully in an answer box. Cheers, –  Matt E Dec 24 '10 at 3:30
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