{"title":"与狄利克雷l函数有关的和的显式公式","authors":"Brahim Mittou","doi":"10.7546/nntdm.2022.28.4.744-748","DOIUrl":null,"url":null,"abstract":"Let $p\\geq3$ be a prime number and let $m, n$ and $l$ be integers with $\\gcd(l,p)=1$. Let $\\chi$ be a Dirichlet character modulo $p$ and $L(s,\\chi)$ be the Dirichlet L-function corresponding to $\\chi$. Explicit formulas for: $$\\dfrac{2}{p-1} \\sum \\limits\\sb{\\underset{\\chi(-1)=+1}{\\chi\\hspace{-0.2cm} \\mod p}} \\chi(l) L(m,\\chi)L(n,\\overline{\\chi}) \\text{ and }\\dfrac{2}{p-1} \\sum \\limits\\sb{\\underset{\\chi(-1)=-1}{\\chi\\hspace{-0.2cm} \\mod p}} \\chi(l) L(m,\\chi)L(n,\\overline{\\chi})$$ are given in this paper by using the properties of character sums and Bernoulli polynomials.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2022-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Explicit formulas for sums related to Dirichlet L-functions\",\"authors\":\"Brahim Mittou\",\"doi\":\"10.7546/nntdm.2022.28.4.744-748\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $p\\\\geq3$ be a prime number and let $m, n$ and $l$ be integers with $\\\\gcd(l,p)=1$. Let $\\\\chi$ be a Dirichlet character modulo $p$ and $L(s,\\\\chi)$ be the Dirichlet L-function corresponding to $\\\\chi$. Explicit formulas for: $$\\\\dfrac{2}{p-1} \\\\sum \\\\limits\\\\sb{\\\\underset{\\\\chi(-1)=+1}{\\\\chi\\\\hspace{-0.2cm} \\\\mod p}} \\\\chi(l) L(m,\\\\chi)L(n,\\\\overline{\\\\chi}) \\\\text{ and }\\\\dfrac{2}{p-1} \\\\sum \\\\limits\\\\sb{\\\\underset{\\\\chi(-1)=-1}{\\\\chi\\\\hspace{-0.2cm} \\\\mod p}} \\\\chi(l) L(m,\\\\chi)L(n,\\\\overline{\\\\chi})$$ are given in this paper by using the properties of character sums and Bernoulli polynomials.\",\"PeriodicalId\":44060,\"journal\":{\"name\":\"Notes on Number Theory and Discrete Mathematics\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2022-11-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Notes on Number Theory and Discrete Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.7546/nntdm.2022.28.4.744-748\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Notes on Number Theory and Discrete Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.7546/nntdm.2022.28.4.744-748","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
Explicit formulas for sums related to Dirichlet L-functions
Let $p\geq3$ be a prime number and let $m, n$ and $l$ be integers with $\gcd(l,p)=1$. Let $\chi$ be a Dirichlet character modulo $p$ and $L(s,\chi)$ be the Dirichlet L-function corresponding to $\chi$. Explicit formulas for: $$\dfrac{2}{p-1} \sum \limits\sb{\underset{\chi(-1)=+1}{\chi\hspace{-0.2cm} \mod p}} \chi(l) L(m,\chi)L(n,\overline{\chi}) \text{ and }\dfrac{2}{p-1} \sum \limits\sb{\underset{\chi(-1)=-1}{\chi\hspace{-0.2cm} \mod p}} \chi(l) L(m,\chi)L(n,\overline{\chi})$$ are given in this paper by using the properties of character sums and Bernoulli polynomials.