On some applications of automorphic forms to number theory

1996/01/01 by Daniel Bump, Solomon Friedberg, Jeffrey Hoffstein · 3 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research

paper · pdf · doi:10.1090/s0273-0979-96-00654-4

openalex publication_date 1996/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/07

Abstract

A basic idea of Dirichlet is to study a collection of interesting quantities <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace a Subscript n Baseline right-brace Subscript n greater-than-or-equal-to 1"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false"></mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:msub> <mml:mo fence="false" stretchy="false"></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\a_n\_{n≥ 1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by means of its Dirichlet series in a complex variable <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w"> <mml:semantics> <mml:mi>w</mml:mi> <mml:annotation encoding="application/x-tex">w</mml:annotation> </mml:semantics> </mml:math> </inline-formula> : <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript n greater-than-or-equal-to 1 Endscripts a Subscript n Baseline n Superscript negative w"> <mml:semantics> <mml:mrow> <mml:munder> <mml:mo> ∑ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:munder> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mi>w</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">∑ _{n≥ 1}a_nn^-w</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In this paper we examine this construction when the quantities <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a Subscript n"> <mml:semantics> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">a_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are themselves infinite series in a second complex variable <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s"> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding="application/x-tex">s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , arising from number theory or representation theory. We survey a body of recent work on such series and present a new conjecture concerning them.

Citations

• Eisenstein series of 1/2-integral weight and the mean value of real DirichletL-series
• Nonvanishing theorems for L-functions of modular forms and their derivatives
• The symmetric cube
• Eisenstein Series on the Metaplectic Group and Nonvanishing Theorems for Automorphic L-Functions and their Derivatives
• On Modular Forms of Half Integral Weight
• Nonvanishing Theorems for Automorphic L-Functions on GL(2)
• Cubic metaplectic forms onGL (3)

Cited by

• Multiple Dirichlet series and moments of zeta and L-functions
• Daniel Bump [wikipedia]
• Jeffrey Hoffstein [wikipedia]