Arithmetic Functions And Their Related Identities Over The Set Of Regular Integers

Number theory functions are one of the key components in the field of math-ematics.It is important to study the arithmetic properties of number theory functions.Among these number theory functions,Ramanujan sums and D-edekind sums are very prominent.Many scholars have made in-depth researches on it and have given many interesting results.In 1960,Cohen gave the defini-tion of unitary divisor and studied some identities related to Ramanujan sums,which laid a theoretical foundation for later scholars.In 2008,Toth gave some equivalent judgments on regular integers,among which the judgment relation between regular integers and unitary divisor provided a powerful tool for later scholars to study regular integers.In 2012,Haukkanen and Toth first proposed a trigonometruc sums on a regular integers set.This arithmetic function is simi-lar to Ramanujan sums.So compare this arithmetic function with Ramanujan sums and describe the Fourier transform of this arithmetic function in detail Arithmetic properties such as Fourier transform,multipliability,mean formula,etc.In 2015,Apostol and Toth studied arithmetic functions similar to the Jordan function on regular integers sets φ(r),and derived the asymptotic for-mula.In 2019,Wang Tingting and Guan Yaliang studied a class of calculation problems related to the sum of Dedekind sums on the set of regular integers and gave an identity.In this paper,starting from the definition of the regular integers of the module n,based on the research of Haukkanen and Toth.The mathematical properties of the function are studied by means of elementary methods and the properties of trigonometruc sums.Firstly,the calculation of an infinite series containing this function is studied by using different methods,then an identity containing this function is given.Secondly,this paper uses analytical methods to study a class of calculation problems related to the sum of Dedekind sums,gives this sums the formula is specifically calculated at some special points.Finally,on the basis of Apostol and Toth research,the asymptotic formula of the function is supplemented,asymptotic formulas for the some inference formulas is given on the regular integer set.