The Distribution Of The Mean Value Of Exponential Function On Zp

The research on the nature of integers occupies a pivotal position in the entire process of number theory development,thus attracting many scholars to carry out extensive and in-depth exploration.At the same time,the primitive root is an important concept in number theory,and it is also of great significance to study its distribution.This paper mainly uses the mean value of the nontrivial upper bound on a class of Kloostermann sums as well as the properties of a new type of exponential sums and exponential sums estimate,combined with elementary methods,to study the distribution of the difference of the exponential function that based on g over Z_p,the mean value of the difference between elements in Z_p and the residue of the exponential function based on g modulo p and an interesting generalization of Lehmer number.we obtain the concrete contents in the following:1.Let p be a prime,and for each integer t>1 with t|p-1,is of multiplicative order t over the residue ring Z_p.The paper studied the mean value of the formula#12 and give asymptotic formula.2.An arbitrary set A(?)Z*p,g is a primitive root of module prime p>3,δ be any real number with 0<δ≤1,Lct#12 For any non-negative integer k,The paper studied the mean value of the formula#12 and give the corresponding asymptotic formula.3.Any real number with 0<δ≤1,g is a primitive root of module prime p then for any non-negative integer k,when the integer a is the generalized Lehmer number,The paper studied the mean value of the formula#12 and give the strong asymptotic formula.