# Do we care about multiple zeta functions?

Coming from a number-theoretic background, I certainly care about $L$-functions and in particular automorphic ones. For automorphic forms on $SL_2(\mathbb{Z}) \backslash SL_2(\mathbb{R})$, $L$-function can be interpreted as the Mellin transform of Fourier expansion at the cusp.

If we look at automorphic forms on $SL_(n,\mathbb{Z}) \backslash SL(n,\mathbb{R})$, we still have a Fourier expansion in $n-1$ variables since it is periodic with respect to the super-diagonal unipotent group. This gives us Fourier coefficients $A(m_1,\cdots,m_{n-1})$. In forming the $L$-functions for $SL(n,\mathbb{R})$, we just look at all the Fourier coefficients $A(m,1,\cdots,1)$ and define $$L(s) = \sum_{m=1}^{\infty} A(m,1,\cdots,1)m^{-s}$$ But the most natural thing to do (comparing to the $SL_2(\mathbb{R})$ case) would be to form the multiple Dirichlet series instead $$\sum_{m_1,\cdots,m_{n-1}} \frac{A(m_1,\cdots,m_{n-1})}{m_1^{s_1} \cdots m_{n-1}^{s_{n-1}}}$$ It certainly seems that people care "less" about this multiple Dirichlet series: in the whole Langlands business we always take about L-parameters and stuff, which seems to imply that we track only the information of $L$-function but not the whole multiple Dirichlet series.

So here are my questions,

• Is there any conceptual reason why we care more about $L$-functions rather than multiple Dirichlet series?
• Of course, Fourier expansion is available when there is a cusp. For cocompact arithmetic quotients of $GL_n(\mathbb{R})$ (or other reductive groups in general), can one similarly define a multiple Dirichlet series that should incorporate the data of $L$-function?

Thank you.

It may help clarify things to work out a specific example, although the OP may know this. In case $n=3$, the double Dirichlet series evaluates as $$\sum_{m,n=1}^{\infty} \frac{A_F(m,n)}{m^{w} n^s} = \frac{L(\overline{F},w) L(F, s)}{\zeta(s+w)}.$$ Here $\overline{F}$ is the contragredient of $F$. This is known as Bump's double Dirichlet series, and this is worked out in Section 6.6 of Goldfeld's book, Automorphic forms and $L$-functions for the group $GL(n,\mathbb{R})$. There are references there for generalizations.

It seems clear (to me) that the $GL_3$ automorphic $L$-function $L(F,s)$ (and its contragredient, and $\zeta$) are the fundamental objects, but also that there are many interesting Dirichlet series that one can construct from an automorphic form.

• Thanks! I will get hold of the book later today to check the references. Meanwhile, is there such a product formula for $GL(n)$-forms? If that's the case I would agree that these multiple Dirichlet series contain no extra information comparing to $L$-functions and that $L$-functions are more fundamental.
– Pig
Feb 2, 2016 at 18:18
• I wasn't able to find an online copy of the paper referred to in Section 6.6 (by Bump and Friedberg), because it is in a conference proceedings. I don't think that it calculates the multiple Dirichlet series you wanted. I'm not aware of any product formula for it, but perhaps nobody has tried to evaluate it. Feb 3, 2016 at 2:22
• Thanks for your answer! By looking at how the identity in $GL(3)$ case is proved I'm pretty convinced that the same method would work in the general case, up to unraveling what the series $\sum_{m=1}^{\infty} \frac{A_F (1,\cdots, 1, m, 1 \cdots, 1)}{m^s}$ means - I'll think about this in more details later. Also, Bump-Friedberg's paper seems to be of a different nature. They were able to write $L(s_1, \pi)L(s_2, \pi, \Lambda^2)$ as some Rankin-Selberg convolution - so they were generalizing the right hand side of the above formula. It does not look like their resulting integral is related..
– Pig
Feb 4, 2016 at 2:07
• to what I am asking.
– Pig
Feb 4, 2016 at 2:07

To address the conceptual question, the $L$-function essentially characterizes the automorphic representation and can be studied locally (associating local $L$-functions to local components of the global representation), so it seems to me there is little need (a priori) for using a more complicated Dirichlet series to study these representations.

On the other hand, multiple Dirichlet series typically do not have Euler products, so do not admit a local-global study, at least in a naive way (though see Bump's survey article). Of course, as Matt indicates, they are useful for studying $L$-functions.

Moreover, from an arithmetic point of view, $L$-functions are naturally related to varieties such as elliptic curves. As far as I know (though I am not an expert on multiple Dirichlet series), there is no direct connection between MDS and counting points on varieties.