Lower order terms for the one-level density of a sympletic family of Hecke L-functions

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F21%3A10438408" target="_blank" >RIV/00216208:11320/21:10438408 - isvavai.cz</a>

Výsledek na webu

<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=V4eZV2as8-" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=V4eZV2as8-</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.jnt.2020.11.022" target="_blank" >10.1016/j.jnt.2020.11.022</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Lower order terms for the one-level density of a sympletic family of Hecke L-functions

Popis výsledku v původním jazyce

In this paper we apply the $L$-function Ratios Conjecture to compute the one-level density for a symplectic family of $L$-functions attached to Hecke characters of infinite order. When the support of the Fourier transform of the corresponding test function $f$ reaches $1$, we observe a transition in the main term, as well as in the lower order term. The transition in the lower order term is in line with behavior recently observed by D. Fiorilli, J. Parks, and A. Södergren in their study of a symplectic family of quadratic Dirichlet $L$-functions. We then directly calculate main and lower order terms for test functions $f$ such that supp($widehat{f}) subset [-alpha,alpha]$ for some $alpha &lt;1$, and observe that this unconditional result is in agreement with the prediction provided by the Ratios Conjecture. As the analytic conductor of these L-functions grow twice as large (on a logarithmic scale) as the cardinality of the family in question, this is the optimal support that can be expected with current methods. Finally as a corollary we deduce that, under GRH, at least 75$%$ of these $L$-functions do not vanish at the central point.

Název v anglickém jazyce

Lower order terms for the one-level density of a sympletic family of Hecke L-functions

Popis výsledku anglicky

In this paper we apply the $L$-function Ratios Conjecture to compute the one-level density for a symplectic family of $L$-functions attached to Hecke characters of infinite order. When the support of the Fourier transform of the corresponding test function $f$ reaches $1$, we observe a transition in the main term, as well as in the lower order term. The transition in the lower order term is in line with behavior recently observed by D. Fiorilli, J. Parks, and A. Södergren in their study of a symplectic family of quadratic Dirichlet $L$-functions. We then directly calculate main and lower order terms for test functions $f$ such that supp($widehat{f}) subset [-alpha,alpha]$ for some $alpha &lt;1$, and observe that this unconditional result is in agreement with the prediction provided by the Ratios Conjecture. As the analytic conductor of these L-functions grow twice as large (on a logarithmic scale) as the cardinality of the family in question, this is the optimal support that can be expected with current methods. Finally as a corollary we deduce that, under GRH, at least 75$%$ of these $L$-functions do not vanish at the central point.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GJ17-04703Y" target="_blank" >GJ17-04703Y: Kvadratické formy a numerační systémy nad číselnými tělesy</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2021

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Journal of Number Theory

ISSN

0022-314X

e-ISSN

—

Svazek periodika

2021

Číslo periodika v rámci svazku

221

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

37

Strana od-do

447-483

Kód UT WoS článku

000613669000017

EID výsledku v databázi Scopus

2-s2.0-85098657530