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I asked this question on m.SE in an attempt to find out the right words to say for these questions I am about to ask.

In his great answer, Matthew Emerton explained that (cuspidal) automorphic L-functions correspond to the Dirichlet series with "nice" properties like having a reflection equation, a meromorphic continuation to the entire complex plane and a suitable analogue of the Riemann hypothesis. This leads me to my first question:

1) Are there Dirichlet series that cannot be classified as (cuspidal) automorphic L-functions, yet still possess a critical line of nontrivial zeroes?

Now, I come to the question I had been meaning to ask here. It is known that functions like Riemann $\zeta$ and the Ramanujan Dirichlet series admit a "Riemann-Siegel" decomposition; that is, letting $\sigma$ denote the position of the "critical line" of the Dirichlet series $g(s)$, they can be expressed as

$$g(\sigma+it)=z(t)\exp(-i\vartheta(t))$$

where $z(t)$ and $\vartheta(t)$ are "Riemann-Siegel" functions corresponding to the Dirichlet series $g(s)$. The value of $z(t)$ is that it eases the task of finding nontrivial zeroes of the corresponding Dirichlet series (essentially helping to verify its corresponding "hypothesis"). My question now is

2) Do all (cuspidal) automorphic L-functions have a "Riemann-Siegel" decomposition? If not, what restrictions are there for them to possess such a decomposition?

My motivation is more of curiosity than anything else. Hopefully this is not too elementary a question!

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2 Answers

In answer to #2, the Riemann-Siegel formula was developed by Siegel based on unpublished posthumous notes of Riemann, the Nachlass. It writes the Hardy function $Z(t)$ as a sum of two finite series of length about $\sqrt{t}$, up to a small error. The two finite series show explicitly the functional equation, i.e one is in $1/2+it$, the other $1/2-it$. Riemann-Siegel replaced earlier $O(t)$ methods of computation such as Euler-Maclaurin summation. Edwards book on Riemann's Zeta Function, reprinted by Dover, is a good source.

Extending this to other Dirichlet series is complicated by the fact that the coefficients are so unruly (the zeta function is the Dirichlet generating function of the constant sequence.) Davies, in "An approximate functional equation for Dirichlet L-functions." Proc. Roy. Soc. Ser. A 284 1965 pp. 224–236. considered the case of Dirichlet $L$-functions and did computations of zeros. Motohashi did $\zeta(s)^2$, which is the Dirichlet generating function of the divisor function. There are some other results, but if constant in the big Oh error in the truncation is not made explicit, they are not useful for computation.

A more modern approach which applies generally to automorphic $L$-functions uses something called an 'Approximate Functional Equation', see section 5.2 of the book Analytic Number Theory by Iwaniec and Kowalski. Iwaniec elsewhere has described this as "a Dirichlet series representation ... tempered by a test function which makes the series rapidly convergent. Formulas of this type are known in the literature as 'approximate functional equations'. [T]his is a somewhat misleading name, because we need exact expressions ... We rather think of these as a kind of Poisson's summation formulas." The test function replaces the sharp cutoff of the a finite series in Riemann-Siegel, which necessitates a corresponding error estimate, with a rapidly decaying test function of the user's choice. I believe this form is due to Lavrik, see MR0188170 (32 #5609). This was first used for computations by PJ Weinberger in "On small zeros of Dirichlet $L$-functions." Math. Comp. 29 (1975), pp. 319–328.

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So, what was so special with, for instance, the Ramanujan Dirichlet series that enabled it to have a(n admittedly rather complicated) "Riemann-Siegel" decomposition? I'd think $\tau(n)$ is an even more unruly function than a Dirichlet character... –  J. M. Dec 29 '10 at 2:52
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Yes, and that result was extremely technically difficult. Also, Gritsenko did the case of the product of two Dirichlet $L$-functions, which in the case of two real characters, are the $L$-functions of genus characters on the class group. Riemann-Siegel can be extended a lot but as I said above, without explicit constants in the big Oh, it's not useful for computation. The 'Approximate Functional Equation' method generalizes Riemann-Siegel, and is much easier. –  Stopple Dec 29 '10 at 23:23
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In answer to #1, I'm not quite sure what you're asking but I'll point out the Epstein zeta function $\zeta_Q(s)=\sum^\prime_{(m,n)}Q(m,n)^{-s}$ attached to, say, a positive definite binary quadratic form $Q(x,y)=ax^2+bxy+cy^2$ has a functional equation under $s\to 1-s$, but except in extraordinary cases (class number 1) they do not possess an Euler product over primes. Setting $z=\frac{-b+i\sqrt{D}}{2a}$ gives a point in the complex upper half plane (these are more or less what Heegener points are) and makes the Epstein zeta function an Eisenstien series. Thus these are automorphic, but not cuspidal.

This has been a area of research on the failure of the Riemann Hypothesis, they are know to have real zeros in $(1/2,1)$ if $\sqrt{d}/a$ is big enough. Up to large height (depending on $d$) the complex zeros are on the critical line.

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Hmm, okay. Is there a Dirichlet series that isn't automorphic but still has "nice" properties, or is that condition absolutely necessary? –  J. M. Dec 30 '10 at 3:59
    
'Nice' is a little vague. From a Dirichlet series $\sum_n a_n n^{-s}$ one can formally construct a function $f(z)=\sum_n a_n\exp(2\pi i nz)$. This $f$ is invariant under $z\to z+1$, i.e automorphic under the action of the matrix $T=\left(\begin{smallmatrix} 1&1\\0&1\end{smallmatrix}\right)$. This is one of the two canonical generators of the modular group $SL(2,\mathbb Z)$. Automorphy under the action of the other, $S=\left(\begin{smallmatrix} 0&1\\-1&0\end{smallmatrix}\right)$, will correspond to the Dirichlet series having a functional equation. –  Stopple Dec 31 '10 at 21:45
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