关于整数矩阵除数函数余项的二次积分均值  

Integral mean square estimation for the error term involving divisor functions of integer matrices

作　　者：于若彤 劳会学 杨晓伟 YU Ruotong;LAO Huixue;YANG Xiaowei(School of Mathematics and Statistics,Shandong Normal University,Jinan 250358,China)

机构地区：[1]山东师范大学数学与统计学院,山东济南250358

出　　处：《纯粹数学与应用数学》2024年第2期203-211,共9页Pure and Applied Mathematics

基　　金：国家自然科学基金(12201363)。

摘　　要：整数矩阵表法个数的渐近分布问题是解析数论中的重要研究课题,受到日益增长的关注.设t_(3)^((2))(n)是整数矩阵环M_(2)(Z)中形式为C=A_(1)A_(2)A_(3)且|C|=n的矩阵表法个数的求和函数,△_(2,3)^(*)(x)是关于t_(3)^((2))(n)的渐近公式中的余项.利用经典的解析方法和黎曼zeta函数的良好性质,本文研究了整数矩阵除数函数t_(3)^((2))(n)在无平方因子数集上的分布问题,并得到了余项△_(2,3)^(*)(x)的二次积分均值的上界估计.The asymptotic behaviour of the number of representations of integer matrices is an important topic in analytic number theory,and has received increasing attention.Let t_(3)^((2))(n) be a summatory function of the number of representations of matrices from the ring of integer matrices M_(2)(Z) in the form C=A_(1)A_(2)A_(3 )with|C|=n.Denote by △_(2,3)^(*)(x) the error term of asymptotic formula related to t_(3)^((2))(n).Applying the classical analytic method and nice properties of the Riemann zeta function,this paper investigates the distribution of divisor functions of integer matrices t_(3)^((2))(n) on the square-free numbers and obtains the upper bounds for the integral mean square of error terms△_(2,3)^(*)(x).

关 键 词：余项 无平方因子数 整数矩阵除数函数 

分 类 号：O156.4[理学—数学]