一类二项指数和的四次幂均值

On the fourth power mean value of one kind two-termexponential sums

指数和C(m,n,r,s;q)的高次幂均值计算与上界估计方面的研究与诸多数论问题联系密切,例如华林问题等。设p为奇素数,关注参数n=1,指数幂r=4,s=2条件下的一类二项指数和的四次幂均值计算问题。利用解析方法,借助Dirichlet特征的奇偶性、正交性及特征和的性质,研究了形如C(m,1,4,2;p)的二项指数和的四次均值计算,给出了在素数p≡3 mod 4情况下上述二项指数和的一个精确的计算公式。同时,对于此类研究内容,该文也提出了一些有待解决的公开问题。

The calculation of the mean value of exponential sums C(m,n,r,s;q)and its upper bound estimation is closely related to many number theory problems,such as Waring’s problem.Let p be an odd prime,this paper focuses on the problem of the calculation of the fourth power mean value of one kind two-term exponential sums when parameter n=1 and exponent r=4,s=2.By utilizing the analytic methods,the odd-even property and orthogonality of Dirichlet character,and the properties of the character sums,we study the calculating problem of the fourth power mean of the two-term exponential sums for the form as C(m,1,4,2;p),and give an exact calculating formula for it when the prime p≡3 mod 4.At the same time,for such research content,some open problems in this field are also proposed.