Multiple Dirichlet Series and Moments of Zeta and L-Functions

By A. Diaconu, Dorian Goldfeld, Jeffrey Hoffstein. Compositio Mathematica, 2001

183 citations recorded by the source corpus.

This paper develops an analytic theory of Dirichlet series in several complex variables which possess sufficiently many functional equations. In the first two sections it is shown how straightforward conjectures about the meromorphic continuation and polar divisors of certain such series imply, as a consequence, precise asymptotics (previously conjectured via random matrix theory) for moments of zeta functions and quadratic L-series. As an application of the theory, in a third section, we obtain the current best known error term for mean values of cubes of cent ral values of Dirichlet L-series. The methods utilized to derive this result are the convexity principle for functions of several complex-variables combined with a knowledge of groups of functional equations for certain multiple Dirichlet series.

Subfields: Algebra and Number Theory, Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/8e6d7e24bee71400fb7207a10a86d499d2370dcd

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In Algebra and Number Theory

In Applied Mathematics

Citations in the corpus

Constructing Weyl group multiple Dirichlet series (2008, methodology): “ies can be used to study the ﬁrst, second and third moments of quadratic Dirichlet L-functions respectively, as in Goldfeld–Hoﬀstein [GH85], Fisher–Friedberg [FF04,FF03] or Diaconu– Goldfeld–Hoﬀstein [DGH03]. The ﬁrst named author used the A5 quadratic series to study mean values of zeta functions o”
Double Dirichlet series and the n-th order twists of Hecke L-series (2003, unknown): “The only potential poles of Z ∗ 1 (s, w; η, ρ) concealed by the Euler products are at � w = 1 ,w � 1 and at � w = 1 −� s ± 1/n, w � 1 − s ± 1/n .I t is then a simple matter to check, as in [ DGH , proof of Prop.”
Moments of the Riemann zeta function (2006, methodology): “[17] Z. Rudnick and K. Soundararajan, Lower bounds for moments of L-functions: symplectic and orthog- onal examples, Multiple Dirichlet Series, Automorphic Forms, and Analytic Number Theory (Editors: Friedberg, Bump, Goldfeld, and Hoffstein), Proc.”
Lower bounds for moments of L-functions: symplectic and orthogonal examples (2006, background): “This problem has been intensively studied in recent years, and thanks to the pioneering work of Keating and Snaith [7], and the subsequent contributions of Conrey, Farmer, Keating, Rubinstein and Snaith [1], and Diaconu, Goldfeld and Hoffstein [ 3 ] there are now well-established conjectures for the”
Autocorrelation of ratios of L-functions (2007, methodology): “More recently, conjectures for all of the main terms of the moments have been found, using heuristics based on the harmonic detector of a family [6] and also using a plausible conjecture for multiple Dirichlet series [13].”
Sharp conditional bounds for moments of the Riemann zeta function (2013, background): “See e.g. the introduction to Conrey and Gonek’s paper [3], and the papers of Conrey and Farmer [2] and of Diaconu, Goldfeld and Hoﬀstein [4], for more discussion about the size of the constants in moment conjectures, from the random matrix and other points of view.”
Weyl group multiple Dirichlet series constructed from quadratic characters (2006, methodology): “t L-functions. More examples of higher rank have also appeared in the literature and have been applied to analytic number theory. The quadratic D4 series was treated by Diaconu, Goldfeld and Hoﬀstein [14], who used it to prove mean value results for sums of cubes of quadratic Dirichlet L-functions. ”
Moments and distribution of central L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L$$\end{document}-values of quadr (2014, background): “There are now precise, and widely believed, conjectured asymptotic formulae for moments in several important families (see [3,9,21,22]), and these conjectures have been verified in a variety of cases (see for example [1,4,31]).”
The fourth moment of quadratic Dirichlet L-functions over function fields (2016, methodology): “An alternative approach to computing moments has been proposed by Diaconu, Goldfeld and Hoffstein [9], using multiple Dirichlet series.”
Conjectures for the integral moments and ratios of L-functions over function fields (2014, methodology): “ (1.23) . Remark 2. Conjecture 3 was originally stated with error term O(|d|− 1 2 +ε), but it appears there are extraneous lower order terms, as ﬁrstly pointed out by Diaconu, Goldfeld and Hoﬀ- stein [DGH], with the remainder term being larger for k≥ 3. This is supported numerically by the computati”