{"title":"关于第 j 次对称幂 L 函数在某些正整数稀疏序列上的傅立叶系数平均行为的说明","authors":"Youjun Wang","doi":"10.21136/cmj.2024.0038-24","DOIUrl":null,"url":null,"abstract":"<p>Let <i>j</i> ⩾ 2 be a given integer. Let <i>H</i><sub><i>k</i></sub>* be the set of all normalized primitive holomorphic cusp forms of even integral weight <i>k</i> ⩾ 2 for the full modulo group SL(2, ℤ). For <i>f</i> ∈ <i>H</i><sub><i>k</i></sub>*, denote by <span>\\({{\\rm{\\lambda }}_{{\\rm{sy}}{{\\rm{m}}^j}{\\kern 1pt} f}}(n)\\)</span> the <i>n</i>th normalized Fourier coefficient of <i>j</i>th symmetric power <i>L</i>-function (<i>L</i>(<i>s</i>, sym<sup><i>j</i></sup><i>f</i>)) attached to <i>f</i>. We are interested in the average behaviour of the sum </p><span>$$\\sum\\limits_{\\scriptstyle n\\, = \\,a_1^2 + a_2^2 + a_3^2 + a_4^2 + a_5^2 + a_6^2x \\atop \\scriptstyle \\,\\,\\,\\,\\,\\,\\,({a_1},{a_2},{a_3},{a_4},{a_5},{a_6}{\\rm{)}} \\in \\,{{\\mathbb{Z}}^6}} {{\\rm{\\lambda }}_{{\\rm{sy}}{{\\rm{m}}^j}\\,f\\left( n \\right),}^2}$$</span><p> where <i>x</i> is sufficiently large, which improves the recent work of A. Sharma and A. Sankaranarayanan (2023).</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A note on average behaviour of the Fourier coefficients of jth symmetric power L-function over certain sparse sequence of positive integers\",\"authors\":\"Youjun Wang\",\"doi\":\"10.21136/cmj.2024.0038-24\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <i>j</i> ⩾ 2 be a given integer. Let <i>H</i><sub><i>k</i></sub>* be the set of all normalized primitive holomorphic cusp forms of even integral weight <i>k</i> ⩾ 2 for the full modulo group SL(2, ℤ). For <i>f</i> ∈ <i>H</i><sub><i>k</i></sub>*, denote by <span>\\\\({{\\\\rm{\\\\lambda }}_{{\\\\rm{sy}}{{\\\\rm{m}}^j}{\\\\kern 1pt} f}}(n)\\\\)</span> the <i>n</i>th normalized Fourier coefficient of <i>j</i>th symmetric power <i>L</i>-function (<i>L</i>(<i>s</i>, sym<sup><i>j</i></sup><i>f</i>)) attached to <i>f</i>. We are interested in the average behaviour of the sum </p><span>$$\\\\sum\\\\limits_{\\\\scriptstyle n\\\\, = \\\\,a_1^2 + a_2^2 + a_3^2 + a_4^2 + a_5^2 + a_6^2x \\\\atop \\\\scriptstyle \\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,({a_1},{a_2},{a_3},{a_4},{a_5},{a_6}{\\\\rm{)}} \\\\in \\\\,{{\\\\mathbb{Z}}^6}} {{\\\\rm{\\\\lambda }}_{{\\\\rm{sy}}{{\\\\rm{m}}^j}\\\\,f\\\\left( n \\\\right),}^2}$$</span><p> where <i>x</i> is sufficiently large, which improves the recent work of A. Sharma and A. Sankaranarayanan (2023).</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-04-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.21136/cmj.2024.0038-24\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.21136/cmj.2024.0038-24","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
设 j ⩾ 2 为给定整数。让 Hk* 是全模态群 SL(2, ℤ)的偶数积分权重 k ⩾ 2 的所有归一化原始全形顶点形式的集合。对于 f∈ Hk*,用 \({{\rm{\lambda }}_{{\rm{sy}}{{\rm{m}}^j}{kern 1pt}} f}}(n)\) 表示连接到 f 的第 j 个对称幂 L 函数 (L(s, symjf)) 的第 n 个归一化傅里叶系数。我们感兴趣的是总和 $$\sum\limits_{\scriptstyle n\, = \、a_1^2 + a_2^2 + a_3^2 + a_4^2 + a_5^2 + a_6^2x \atop \scriptstyle \,\,\,({a_1},{a_2},{a_3},{a_4},{a_5},{a_6}{\rm{)}}。\in\,{{mathbb{Z}}^6}}{{\rm{lambda }}_{{\rm{sy}}{{\rm{m}}^j}}\,f\left( n \right),}^2}$$ 其中 x 足够大,这改进了 A. Sharma 和 A. Sankaranarayanan (2023) 最近的工作。
A note on average behaviour of the Fourier coefficients of jth symmetric power L-function over certain sparse sequence of positive integers
Let j ⩾ 2 be a given integer. Let Hk* be the set of all normalized primitive holomorphic cusp forms of even integral weight k ⩾ 2 for the full modulo group SL(2, ℤ). For f ∈ Hk*, denote by \({{\rm{\lambda }}_{{\rm{sy}}{{\rm{m}}^j}{\kern 1pt} f}}(n)\) the nth normalized Fourier coefficient of jth symmetric power L-function (L(s, symjf)) attached to f. We are interested in the average behaviour of the sum