{"title":"Asymptotics for the second moment of the Dirichlet coefficients of symmetric power L-functions","authors":"Xue Han, Huafeng Liu","doi":"10.1007/s10986-024-09636-0","DOIUrl":null,"url":null,"abstract":"<p>Let <i>m</i> ≥ 2 be an integer. Let <i>f</i> be a holomorphic Hecke eigenform of even weight <i>k</i> for the full modular group <i>SL</i>(2, ℤ). Denote by <i>λ</i><sub>Sym</sub><sup><i>m</i></sup> <sub><i>f</i></sub> (<i>n</i>) the <i>n</i>th normalized Dirichlet coefficient of the corresponding symmetric power <i>L</i>-function <i>L</i>(<i>s</i>, Sym<sup><i>m</i></sup><i> f</i>) related to <i>f</i>. In this paper, we study the average behavior of the second moment of the Dirichlet coefficients <i>λ</i><sub>Sym</sub><sup><i>m</i></sup> <sub><i>f</i></sub> (<i>n</i>) and establish its asymptotic formula.</p>","PeriodicalId":51108,"journal":{"name":"Lithuanian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Lithuanian Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10986-024-09636-0","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let m ≥ 2 be an integer. Let f be a holomorphic Hecke eigenform of even weight k for the full modular group SL(2, ℤ). Denote by λSymmf (n) the nth normalized Dirichlet coefficient of the corresponding symmetric power L-function L(s, Symm f) related to f. In this paper, we study the average behavior of the second moment of the Dirichlet coefficients λSymmf (n) and establish its asymptotic formula.
设 m ≥ 2 为整数。设 f 是全模态群 SL(2, ℤ) 偶数权 k 的全形赫克特征形式。用 λSymm f (n) 表示与 f 有关的相应对称幂 L 函数 L(s, Symm f) 的第 n 个归一化 Dirichlet 系数。本文将研究 Dirichlet 系数 λSymm f (n) 的第二矩的平均行为，并建立其渐近公式。
期刊介绍：
The Lithuanian Mathematical Journal publishes high-quality original papers mainly in pure mathematics. This multidisciplinary quarterly provides mathematicians and researchers in other areas of science with a peer-reviewed forum for the exchange of vital ideas in the field of mathematics.
The scope of the journal includes but is not limited to:
Probability theory and statistics;
Differential equations (theory and numerical methods);
Number theory;
Financial and actuarial mathematics, econometrics.
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