The authors call \[ \sum_{m_1=1}^\infty\cdots\sum_{m_n=1}^\infty \int_0^\infty\cdots\int_0^\infty a(m_1,\ldots,m_n, t_1,\ldots,t_\ell)t_1^{-w_1} \cdots t_\ell^{-w_\ell}\,dt_1\cdots\,dt_\ell \]
a multiple Dirichlet series, provided that \(a(m_1,\ldots,m_n,t_1,\ldots,t_\ell)\) is a complex-valued smooth function. One of the simplest examples is \(\sum_d L(s,\chi_d)| d| ^{-w}\), where the sum ranges over fundamental discriminants of quadratic fields, \(\chi_d\) is the quadratic character associated to these fields, and \(L(s,\chi_d) =\sum_{n=1}^\infty \chi_d(n)n^{-s}\) is the classical Dirichlet \(L\)-function. As a consequence of their theory they show that, with explicit constants \(a_3, d_i\),
\[ \sum_{| d| \leq x} L(1/2, \chi_d)^3 = {6\over\pi^2}a_3\cdot{1\over2880}x\log^6x +\sum_{i=0}^5d_ix\log^ix + O_\varepsilon(x^{\theta+\varepsilon}),\tag{1} \]
where \(\theta = {1\over36}(47-\sqrt{265}) = 0.853366\ldots\,\).
This improves the result of K. Soundararajan [Ann. Math. (2) 152, 447–488 (2000; Zbl 1036.11042)], who obtained (1) with \(\theta = 11/12\). Besides (1), they obtain also the asymptotic formula for the smoothed version of the sum in question (with the factor \((1-| d| /x)\) inserted), for which they obtain the exponent \(4/5+\varepsilon\), the limit of their method. An important class of multiple Dirichlet series (of \(n\) complex variables) are those which have meromorphic continuation to \(\mathbb C^n\) and satisfy also a group of functional equations (similar to the functional equation \(\zeta(s) = \chi(s)\zeta(1-s)\) for the Riemann zeta-function). These are the so-called perfect multiple Dirichlet series, and the case of cubic moments is the last instance when (for the family of quadratic \(L\)-functions) these series are understood completely. The multiple Dirichlet series \[ Z(s_1,s_2,\cdots,s_m;w) = \sum_d L(s_1,\chi_d)L(s_2,\chi_d)\cdots L(s_m,\chi_d)| d| ^{-w} \tag{2} \]
is extensively discussed in Section 3. The results on \(Z(s_1,s_2,\cdots,s_m;w)\) are applied in Section 4 to cubic moments of \(L\)-series, culminating in the proof of (1). The methods that are used in the authors’ extensive investigations include the convexity principle for functions of several complex variables and the knowledge of groups of functional equations for certain multiple Dirichlet series.
The major objective of the paper is, at least conjecturally, to pass the barrier \(m\geq4\). It is shown that if the multiple Dirichlet series (2) has suitable meromorphic continuation to a tube containg the point \(({1\over2},\cdots, {1\over2},1)\), then
\[ \sum_{| d| \leq x}L^m(1/2,\chi_d) \sim {6\over\pi^2}a_m\prod_{\ell=1}^m{\ell!\over(2\ell)!}x (\log x)^{m(m+1)/2}\qquad(x\to\infty), \] where \(a_m\) is precisely the constant predicted by J. B. Conrey and D. W. Farmer [Int. Math. Res. Not. 17, 883–908 (2000; Zbl 1035.11038)] who obtained their results by using methods from random matrix theory. In Section 2, the authors develop the theory of multiple Dirichlet series
\[ Z(s_1,\cdots,s_{2m},w) = \int_1^\infty \zeta(s_1+it)\cdots \zeta(s_m+it)\zeta(s_{m+1}-it) \cdots\zeta(s_{2m}-it)t^{-w}\,\text{ d}t\tag{3} \]
for the Riemann zeta-function. It is shown that (3) has meromorphic continuation (as a function of \(2m+1\) complex variables) slightly beyond the region of absolute convergence, with a polar divisor at \(w=1\). It is also shown that (3) satisfies certain quasi-functional equations, which are used to obtain meromorphic continuation to an even larger region.
Under the assumption that \(Z({1\over2},\cdots,{1\over2},w)\) has holomorphic continuation to the region \(\operatorname{Re}\, w \geq 1\) (except for the multiple pole at \(w=1\)), the authors derive the conjecture on the moments of the zeta-function on the critical line (see the papers of J. B. Conrey and D. W. Farmer [loc. cit.], J. B. Conrey and S. M. Gonek [Duke Math. J. 107, 577–604 (2001; Zbl 1006.11048)] and J. P. Keating and N. C. Snaith [Comment. Math. Phys. 214, 91–110 (2000; Zbl 1051.11048)]). This is another instance of the power of the authors’ methods involving multiple Dirichlet series.
Reviewer’s remark: In the formulation of Theorem 1.1 on p. 299 the exponent of \(x\) in the error term should be \(4/5+\varepsilon\). In Proposition 2.3 on p. 308 in \(R_r\) there should be \(\prod_p(1-1/p)^{r(2m-r)}\) instead of \(\prod_p(1-1/p)^{m^2}\).