Generalizations Of The Menon-sury's Identity

The main aim of this thesis is to generalize the Menon-Sury's identity in different ways.The classical Menon-Sury's identity reads as follows:???Zn*b1,…,br?Zn gcd(a-1,b1,b2,…,br,n)=?(n)?r(n),where n is a positive integer,Zn*is the group of units of the ring Zn=Z/nZ,gcd(,)de-notes the greatest common divisor,? is the Euler's totient function and ?r(n)=?d|n dr.Our first generalization is to reprove a known result on the Menon-Sury's identity with a Dirichlet character by using the orthogonality of the Dirichlet character and some elementary calculations.Our second generalization is to obtain the Menon-Sury's identity with several Dirichlet characters by using the filtrations of the ring Zn and its unit group Zn*.Based on the second generalization,we also compute the Menon-Sury's identity with several Dirichlet characters and several additive characters.Our last generalization is to consider the situation of Fq[T],the polynomial ring,over a finite field.In which,we obtain a Menon-Sury's identity with several Dirichlet characters and several additive characters for the general arithmetic functions.