MULTIPLE DIRICHLET SERIES AND AUTOMORPHIC FORMS

9

(3) Suzuki [57] and Banks-Bump-Lieman [3], generalizing earlier work of Bump

and Hoffstein [18], showed that there is a metaplectic Eisenstein series on the n-fold

cover of GL(n) (induced from the theta function on the n-fold cover of GL(n — 1))

whose Whittaker coefficients are n-th order twists of a given GL(1) L-series. An

integral transformation yields a sum of twists of GL(1):

where £ is on GL(1) and is fixed. One should then be able to control such sums;

however, the technical difficulties are substantial, as discussed in paragraph 2.3.2

below. In Farmer, Hoffstein, and Lieman [29], mean value results for cubic L-

series were obtained by this approach. (This series has been studied by Friedberg,

Hoffstein, and Lieman [34], using a different method that is explained in Section

4.1 below.)

(4) Similarly, working with n-th order twists, A. Diaconu [26] studied

^\L{s^)\2

This can be obtained from a Rankin-Selberg integral convolution of the metaplectic

Eisenstein series on the n-fold cover of GL(n) described above. Once again, Diaconu

used a different strategy to study this integral, as we shall explain.

2.3.2. Obstructions. In the above paragraph, we described a number of multiple

Dirichlet series that arose as Rankin-Selberg type integrals. Unfortunately, it turns

out to be rather difficult to study the series using such constructions. The following

obstructions arise:

(1) Truncation: The integrals involving Eisenstein series need to be truncated

or otherwise renormalized in order to converge. This can be handled in

principle via the general theory of Arthur [1]. It is, however, complicated

to do in the situations above;

(2) Bad finite primes: Bad finite primes are difficult to handle in Rankin-

Selberg type integrals, unlike the Langlands-Shahidi method (for the lat-

ter, see [53] and the references there). This is particularly true in the case

of integrals involving metaplectic automorphic forms, where the primes

dividing n present additional complications;

(3) Archimedean places: Integrals of archimedean Whittaker functions arise

in the integrals. But the general theory of such integrals is not fully

developed. This is possibly the most serious obstruction to this approach.

Since many properties of L-functions are already known, one might hope that

one can write down and study multiple Dirichlet series without needing to employ

Rankin-Selberg integrals. Remarkably, this is possible in many cases, and it is one

main goal of this paper, and succeeding papers, to explain how. However, we note

that information obtained from metaplectic Eisenstein series does play a key role

in the study of higher twists, as we shall explain also explain in Section 4 below.