The Study Of L-function Square Mean Value And Generalized Exponent And Mean Value

This dissertation focuses on the fourth power mean of the generalized two-term exponential sums G（m,n,k,h,χ;q）and the mean square value of Dirichlet L-functions.It is well known that exponential sums and Dirichlet L-functions are very important for Analytic Number Theory.To study properties of mean value of Dirichlet L-functions is very important for studying some analytical properties of the complex valued functions L（s,χ）.Simultaneously,as an application of exponential sums,this dissertation also focuses on the number of incongruent solutions of some diagonal congruence equations of high degree.More precisely,the main contents and the main results of this dissertation are summarized as follows:1.The generalized two-term exponential sums G（m,n,k,h,χ;q）are one of the most important sums of exponential sums.The generalized two-term exponential sums |G（m,n,k,h,χ;q）| have a lot of good properties,for example |G（m,n,k,h,χ;q）|is a multiplicative function of q.In the second chapter,the fourth power mean of the generalized two-term exponential sums G（m,n,3,1,χ;p）are studied.When the integer n and the odd prime number p satisfy（3,p-1）=1,（n,p）=1,we give an exact formula for the sum（?）.In this research process,we use some analytic methods involving the classical manipulation of exponential sums and some properties of the classical Gauss sums.Certainly we also use a few basic results and conclusions from elementary number theory,including the orthogonality of characters mod p,some properties of reduced residue system mod p and so on.2.In the third chapter,the mean square value of Dirichlet L-functions with the weight of quadratic Gauss sums are studied.If the odd prime number p and two integers α,n satisfy p≡3 mod 4,（n,p）=1,α≥ 2,for the following sums:（?）we use the analytic methods and the properties of the classical Gauss sums to obtain two explicit formulas expressing above-mentioned sums where the summation（?）*is over all odd primitive characters χ mod pα.These results can help us to understand the deeply relationship between the Dirichlet L-function L（s,χ）and the generalized quadratic Gauss sums G（n,χ2;q）.3.In the fourth chapter,we study one kind of special sums（?） where χ6 is an odd character mod 6.If the integer q satisfies（q,6）=1,{d:d ∈N+,d|q}（?）{6h+1:h∈Z)or the integer q is an odd prime which satisfies p=-1 mod 6,by using the reciprocal formula of Dedekind sums and Mobius inversion formula,we respectively give an exact computational formula for（?）.In this chapter,we actually study the computational problem of one kind special mean square value of Dirichlet L-functions.4.In the fifth chapter,as an application of exponential sums,we mainly study the following diagonal congruence equations:（x1,...,xs）∈（Zn*）s:θ1x12+…θsxs2+θ≡0 mod n（s≥2）,where θ1,...,θs,θ∈Z,n ∈ N+.Firstly,we solve a conjecture which is proposed by Quanhui Yang and Min Tang.We use congruence conclusions and a series of computational skills of exponential sums to obtain an exact formula expressing the number of solutions of the following diagonal congruence equation:（x,y）∈（Z_n*）²:θ₁x²+θ₂y²+θ₃≡0 mod n.Secondly,through the method of partition of a set and some expressions for （?）,we obtain a series of formulas for the number of solutions of the diagonal congruence equations:（x,y,z）∈(Z*_p^α)3:θ’₁x²+θ’₂y²+θ’₃z²+θ’₄=0 mod p^α.These results establish the relationship between the Legendre symbol and the number of incongruent solutions of some diagonal congruence equations.