Mean Value Problems Of Divisor Function And Fourier Coefficients Of Automorphic L-Function With Quadratic Forms Variables

The quadratic forms g(m1,m2):=m12+m22, g(m1,m2,m3):=m12+m22+m32, g(m1,m2,m3,m4):=m12+m22+m32+m42, are important in number theory. Some authors studied different kinds of quadratic forms by different methods.In the case of binary quadratic form, Gang Yu [41] studied mean value prob-lems about divisor function and obtainedIn the case of ternary quadratic form, the distribution of primes related to inte-gral vectors in sphere is important in analytic number theory. Vinogradov [39] and Chen [3] established independently that Afterwards, the exponent of x in the error term was refined by Chamizo and Iwaniec [2] to 29/44. Heath-Brown [12] further improved this result to 21/32. Friedlander and Iwaniec [6] considered this problem related to primes and proved that Guo and Zhai [9] showed that for any A> 0, where C3 and I3 are the singular series and singular integral. It follows from the above result that π3(x)=12C3I3∫2xt1/2/logt dt+O(x3/2log-A x). Calderon and Velasco [1] studied sums of the divisor of the ternary quadratic form and concluded that Afterwards, Guo and Zhai [9] improved this result to S(x)=2C1I1x3 log x+(C1I2+C2I1)x3+O(x8/3+ε), where Ci,Ii(i=1,2) are constants. Zhao [43] refined the error term of the above formula to x2 log7 x.In this paper, we firstly consider some problems with quaternary quadratic form and almost equal variables. Then we study the mean value problems about the ternary quadratic forms connected with the Fourier coefficients of automorphic L-function. The following are our results of first part. Theorem 1 Define Then for x≥2,we have S(x)=2K1L1x4 log x+(K1L2+K2L1)x4+O(x7/2+ε), where L1:=∫-∞∞I1(λ)dλ,L2∫-∞∞I2(λ)dλ, I1(λ)=(∫01e(u2λ)du)4∫04e(-uλ)du. I2(λ)=(∫01e(u2λ)du)4∫04e(-uλ)log udu. Theorem 2 Define Then for any fixed constant A>0,we have πΛ(x)=16K3L3x2+O(x2 log-Ax)(x≥2), where L3:=∫-∞∞I3(λ)dλ, I3(λ):=(∫01e(u2λ)du)4∫01e(-uλ)du.To deal with the almost equal problems about Theorem 1 and 2,we define where y=xθ+ε with 0<θ<1.We will establish the following:Theorem 3 For θ=1/3+ε,we haveS(x,y)=2ζ(2)/7ζ(3)L1(x,y)+4ζ(2)/7ζ(3)(γ+12/7+2ζ(2)/ζ(2)-2ζ(3)/ζ(3))L2(x,y)+O(y4-ε), where and satisfy L1(x,y)(?)y4 log y,L2(x,y)(?)y4.Theorem 4 For θ=4/5,we haveS(x,y)=2ζ(2)/7ζ(3)L1(x,y)+4ζ(2)/7ζ(3)(γ+12/7+2ζ(2)/ζ(2)-2ζ(3)/ζ(3))L2(x,y)+O(y7/2+ε), where Li(x,y)are defined as Theorem 3. We next consider the problem related to primes: with y=xδ(0<δ≤1).We will establish the following:Theorem 5 Suppose that δ≥15/23+2ε.For any A>0, πΛ(x,y)=16(?)y4+O(y4L-A), where (?) is the singular series defined as in (1.1).Theorem 6 Suppose that y=xδ with δ satisfying 15/23+2ε≤θ≤1.Define For any A>0,we have π4(x,y)=1/16(?)+O(y4L-A),In the last of this paper, we study the mean value problems about the ternary quadratic forms connected with the Fourier coefficients of automorphic L-function. Let λ(n) and a(n) denotes the normalized Fourier coefficients of Maass cusp forms and holomorphic cusp forms respectively, we get the following results.Theorem 7 Define We have πλ，Λ(x)＝O(x3/2 logx x), where c> 0 is a fixed constant.Theorem 8 Define We have πa,Λ(x)＝O(x3/2 logc’ x), where c’> 0 is a fixed constant.