Some Identities Involving Character Sums

In analytic number theory,the properties and applications of Gauss sums and Dirichlet characters have always been an important part of the research,and many scholars have made many meaningful achievements in these fields.The main purpose of this paper is to use the analytic methods and the properties of Gauss sums and residue classes to study the special sum of the Dirichlet characters under some conditions,as follows:where p and q be two odd primes with q|p-1,? be any non-principal character mod p with ?p ? ?0,r is not a gth residue mod p with(r,p)= 1.First of all,Let q = 3 with 3|p-1,then we have the identity,Then,Let g = 5 with 5|p-1,then we have the identity,Finally,the generalization results are obtained.When p and q be two odd primes with g|q-1,we have the identity,...