One-level density estimates for Dirichlet L-functions with extended support

Sary Drappeau; Kyle Pratt; Maksym Radziwiłł

doi:10.2140/ant.2023.17.805

One-level density estimates for Dirichlet L-functions with extended support

Sary Drappeau, Kyle Pratt, Maksym Radziwiłł

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We estimate the 1-level density of low-lying zeros of L(s, χ) with χ ranging over primitive Dirichlet^{[ ]} characters of conductor in (Formula presented) and for test functions whose Fourier transform is supported in (Formula presented) Previously, any extension of the support past the range (−2, 2) was only known conditionally on deep conjectures about the distribution of primes in arithmetic progressions, beyond the reach of the generalized Riemann hypothesis (e.g., Montgomery’s conjecture). Our work provides the first example of a family of L-functions in which the support is unconditionally extended past the “diagonal range” that follows from a straightforward application of the underlying trace formula (in this case orthogonality of characters). We also highlight consequences for nonvanishing of L(s, χ).

Original language	English (US)
Pages (from-to)	805-830
Number of pages	26
Journal	Algebra and Number Theory
Volume	17
Issue number	4
DOIs	https://doi.org/10.2140/ant.2023.17.805
State	Published - 2023

Funding

Part of this work was conducted while Pratt was supported by the National Science Foundation Graduate Research Program under grant number DGE-1144245. Radziwiłł acknowledges the support of a Sloan fellowship and NSF grant DMS-1902063. The authors thank the referee for helpful remarks, and Jared Lichtman for helpful discussions on Proposition 6.

Keywords

Dirichlet L-functions
arithmetic progressions
dispersion method
nonvanishing
one-level density
primes

ASJC Scopus subject areas

Algebra and Number Theory

Access to Document

10.2140/ant.2023.17.805

Cite this

@article{39329b3e99394f8c84cc9943c3d0d30a,

title = "One-level density estimates for Dirichlet L-functions with extended support",

abstract = "We estimate the 1-level density of low-lying zeros of L(s, χ) with χ ranging over primitive Dirichlet[ ] characters of conductor in (Formula presented) and for test functions whose Fourier transform is supported in (Formula presented) Previously, any extension of the support past the range (−2, 2) was only known conditionally on deep conjectures about the distribution of primes in arithmetic progressions, beyond the reach of the generalized Riemann hypothesis (e.g., Montgomery{\textquoteright}s conjecture). Our work provides the first example of a family of L-functions in which the support is unconditionally extended past the “diagonal range” that follows from a straightforward application of the underlying trace formula (in this case orthogonality of characters). We also highlight consequences for nonvanishing of L(s, χ).",

keywords = "Dirichlet L-functions, arithmetic progressions, dispersion method, nonvanishing, one-level density, primes",

author = "Sary Drappeau and Kyle Pratt and Maksym Radziwi{\l}{\l}",

note = "Publisher Copyright: {\textcopyright} 2023 MSP.",

year = "2023",

doi = "10.2140/ant.2023.17.805",

language = "English (US)",

volume = "17",

pages = "805--830",

journal = "Algebra and Number Theory",

issn = "1937-0652",

publisher = "Mathematical Sciences Publishers",

number = "4",

}

TY - JOUR

T1 - One-level density estimates for Dirichlet L-functions with extended support

AU - Drappeau, Sary

AU - Pratt, Kyle

AU - Radziwiłł, Maksym

PY - 2023

Y1 - 2023

N2 - We estimate the 1-level density of low-lying zeros of L(s, χ) with χ ranging over primitive Dirichlet[ ] characters of conductor in (Formula presented) and for test functions whose Fourier transform is supported in (Formula presented) Previously, any extension of the support past the range (−2, 2) was only known conditionally on deep conjectures about the distribution of primes in arithmetic progressions, beyond the reach of the generalized Riemann hypothesis (e.g., Montgomery’s conjecture). Our work provides the first example of a family of L-functions in which the support is unconditionally extended past the “diagonal range” that follows from a straightforward application of the underlying trace formula (in this case orthogonality of characters). We also highlight consequences for nonvanishing of L(s, χ).

AB - We estimate the 1-level density of low-lying zeros of L(s, χ) with χ ranging over primitive Dirichlet[ ] characters of conductor in (Formula presented) and for test functions whose Fourier transform is supported in (Formula presented) Previously, any extension of the support past the range (−2, 2) was only known conditionally on deep conjectures about the distribution of primes in arithmetic progressions, beyond the reach of the generalized Riemann hypothesis (e.g., Montgomery’s conjecture). Our work provides the first example of a family of L-functions in which the support is unconditionally extended past the “diagonal range” that follows from a straightforward application of the underlying trace formula (in this case orthogonality of characters). We also highlight consequences for nonvanishing of L(s, χ).

KW - Dirichlet L-functions

KW - arithmetic progressions

KW - dispersion method

KW - nonvanishing

KW - one-level density

KW - primes

UR - http://www.scopus.com/inward/record.url?scp=85159838803&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85159838803&partnerID=8YFLogxK

U2 - 10.2140/ant.2023.17.805

DO - 10.2140/ant.2023.17.805

M3 - Article

AN - SCOPUS:85159838803

SN - 1937-0652

VL - 17

SP - 805

EP - 830

JO - Algebra and Number Theory

JF - Algebra and Number Theory

IS - 4

ER -

One-level density estimates for Dirichlet L-functions with extended support

Abstract

Funding

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this