Estimate For Character Sums Over Special Sets

Character sums,one of the important research subjects in analytic num-ber theory,is play a very important significance in the development of analytic number theory.The research of Character sums has been well treated.Estimate of character sums plays an important role and it has intimate relationship with many famous problems of number theory.The estimate of various upper bounds of character sums has received the widespread attention of scholars both home and abroad.Many scholars have devoted to the improvement and optimization of the estimation.Researching the upper bound estimate of character sums on different sets plays an important role in revealing the distribution of character sums.This paper studies the distribution of character sums on some special sets of number and get a sharp upper bound estimate.The study mainly uses the related properties of character sums and other arithmetical functions,meanwhile combines the method of elementary number theory or some elementary methods in related literatures.The main results as following:1.Let l>2 be an integer and q be an square-free with q ? 3,x denote the non-principal character modulo q,for any integer 3 ? H ? q,let S(H,q)={a?Z|1?a?q,(a,q)= 1,|a-(al)q|? H},then we have(?)where w(q)denotes the number of distinct prime factors of q.2.Let q be prime,m ? 2 be an integer,x denote the odd character modulo p,let L(p)= {a?Z|1?a?p,(a,p)= 1,2|((am)p+a)},we have(?)3.Let q? 3 be an odd,x denote the non-principal character modulo q,for any integer 3 ? H ?q,let F(H,q)= {a? Z|(a,q)= 1,1 ?a,b ?q,ab?1(mod q),|a-b| ? H,2 (?)(a+b)},we have...