Some Research On Diophantine Problem And The Mean Square Value Of L-functions

Diophantine problem and the mean square value of L-function are important research in number theory.Where diophantine equation and diophantine approximation are two main research contents of diophantine problem.This paper,the recursive sequence method of elementary number theory is used to solve a high order indefinite equation.Using analytic number theory are studies the mixed power diophantine approximation problems with prime variables and the mean square value problems of L-functions.The main results are as follows:1.It is proved that the diophantine equation 5x(x+1)(x+2)(x+3)=18y(y+1)(y+2)(y+3)has only four non-trivial solutions(x,y)=(6,4),(-9,4),(6,-7),(-9,-7),and its all integer solutions are(x,y)=(0,0),(0,-1),(0,-2),(0,-3),(-1,0),(-1,-1),(-1,-2),(-1,-3),(-2,0),(-2,-1),(-2,-2),(-2,-3),(-3,0),(-3,-1),(-3,-2),(-3,-3),(6,4),(-9,4),(6,-7),(-9,-7).2.It is proved that for any given positive integerk≥ 3,ε>0 ands(k)relies on k,the number of v∈V with v≤V for which |λ1p12+λ2p22+λ3p33+λ4p4k-v|<v-δ without solution in primes p1,p2,p3,p4 does not exceed O(Xσ+2δ+ε),where σ=7/8for 3≤k≤6,σ=29/32 for 7≤k≤12,σ=11/12 for 13 ≤ k≤15,and σ=15/16-1/(16s(k))for k≥16.3.The computational problems of one kind mean square value of Dirichlet L-functions are studied,several exact computational formulas are given for(?)|L(1,xλ)|2,where the summation over all even characters X mod q,λ is a fixed odd character mod r with q,r≥ 3and(r,q)=1.