A Class Of Generalized Euler Functions And The Related Equations

In 2007,Cai,etc,defined the generalized Euler function ?e(n)of positive integer n(e? Z+).and gave an accurate calculation formula for ?e(n)(e = 3,4,6).This thesis further studies the accurate calculation formula of the generalized Euler function.Based on the definitions of the generalized Euler function and Mobius function,the calculation formula of?5(n)for several classes of positive integers is obtained,and then the parity of ?5(n)is studied.Finally,by using elementary methods and techniques,the equation ?e(n)= n/d(e = 1,2,3,4)is studied,the positive integer solutions for the above equation are given when e = 1,2,4 and the necessary condition for the existence of the solutions is also obtained for the case e = 3.