Multiple dirichlet series and moments of zeta and L-functions

Adrian Diaconu, Dorian Goldfeld, Jeffrey Hoffstein

Research output: Contribution to journalArticle

57 Citations (Scopus)

Abstract

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.

Original languageEnglish (US)
Pages (from-to)297-360
Number of pages64
JournalCompositio Mathematica
Volume139
Issue number3
DOIs
StatePublished - Dec 1 2003

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Dirichlet Series
L-function
Riemann zeta function
Several Complex Variables
Moment
Functional equation
Series
Precise Asymptotics
Random Matrix Theory
Meromorphic
Error term
Mean Value
Divisor
Regular hexahedron
Dirichlet
Continuation
Convexity
Imply

Keywords

  • L-functions
  • Moments
  • Multiple dirichlet series
  • Twists
  • Zeta functions

Cite this

Multiple dirichlet series and moments of zeta and L-functions. / Diaconu, Adrian; Goldfeld, Dorian; Hoffstein, Jeffrey.

In: Compositio Mathematica, Vol. 139, No. 3, 01.12.2003, p. 297-360.

Research output: Contribution to journalArticle

Diaconu, Adrian ; Goldfeld, Dorian ; Hoffstein, Jeffrey. / Multiple dirichlet series and moments of zeta and L-functions. In: Compositio Mathematica. 2003 ; Vol. 139, No. 3. pp. 297-360.
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