On The Mean Value Of The Gcd-sum Function Over The Set Of Regular Integers

The mean value problem of arithmetical functions is very important part in the field of mathematics,which is of great significance for studying the properties of functions.Many scholars have given the mean value formula of the classical arithmetic function.In the last ten years,many experts and scholars have studied the mean value formula of the new arithmetical function and the classical arithmetic function involving some special sets.In 1972,Professor Kaplansky gives a necessary and sufficient condition of regular integers modulo 9)for the further study of the properties of regular integers modulo 9)and mean value formulas of multiplicative arithmetical functions involving sets of regular integers.For example,Professor T?oth gives the related properties and equivalent propositions of regular integers modulo 9),the formula of mean value and the correlation properties of the gcd-sum function involving the set of regular integers.In this paper,the gcd-sum function involving the set of regular integers modulo 9)and the mean value formula of the alternating sums of concerning multiplicative arithmetic function are studied in depth,taking the regular integer and the multiplicative arithmetical function as the research object.firstly,Kloostermann's sum and the estimate of trigonometric sum are used to study the computation problems of a class of summation involving regular number sets,and a strong asymptotic formula is given.Secondly,by using the Dirichlet product of arithmetical functions and the properties of multiplicative arithmetical functions,the gcd-sum function involving the set of regular integers modulo 9)are further extended to the generalized gcd-sum function involving the set of regular integers modulo 9).The formula of the mean value of the generalized gcd-sum function involving the set of regular integers modulo 9)is also calculated.Finally,on the basis of the study of Bordell'e s,Cloitre and T?oth,by using Dirichlet product of arithmetical function,the properties of the coefficients of Bell series of 1)modulo 2,we established the mean value formula of the alternating function of the gcd-sum function involving the set of regular integers modulo9).Furthermore,the mean value formula of multiplicative arithmetical function are used,the mean value formula of alternating sum of the corresponding multiplicative arithmetical functions are given.