| Many scholars are interested in researching the properties of the ternary quadratic form m12+m22+m32. Suppose that x is a large positive real number. In 1963, Vino-gradov [19] and Chen [4] independently studied the number of lattice points in the 3-dimensional ball u12 + u22 +u3?x and showed the asymptotic formulaSubsequently, the exponent 2/3 in the above error term was improved to 29/44 by Chamizo and Iwaniec [21, and to 21/32 by Heath-Brown [8].Recently, several authors studied some problems connected with the ternary quadratic form by different methods. Let ?3(x) denote the number of integer points(m1, m2,m3)?Z3 with m12+m22+m32=p ?x.Friedlander and Iwaniec [5] proved thatwhich can be viewed as a generalization of the prime number theorem. Let ?(n)stands for von Mangoldt function, i.e.Guo and Zhai [6] studied the asymptotic behavior of sum S(x) :=? ?(m12+m22+m32)m12+m22+m32?x and obtained S(x) = 8C3I3x3/2 + O(x3/2 log-A x),where A > 0 is a fixed constant,andLet f be a Maass cusp form for SL2(Z) with Laplace eigenvalue 1/4+ v2. Nor-malizing f with the first coefficient being 1, the Fourier expansion of f becomes where Ks(y)is the K-Bessel function with s=1/2+it. Then the L-function attached,to f is defned as where Ks(y) is the K-Bessel function with s =1/2+it. Then the L-function attached to f is defined asThe series above converge absolutely for Re s > 1 (see [12]). Applying the Theorem of K. Chandrasekhaxan and R. Narasimhan[3], we can deduce that?|?(n)|2<<x (1.1)n?x By this bound and Cauchy's inequality, we have?|?A(n)|<<(?|?(n)|2)1/2(?12)1/2<<x (1.2)n<x n<x n<x These two bounds (1.1) and (1.2) will be used in our proof in the sequel.In 2015, Hu [9] studied the asymptotic behavior of sum T(x):=? ?{m12 + m22 + m23)?(m12+m22+m32)m12+m22+m31?x and proved T(x)<<3/2 logcx,where c > 0 is a constant. In 2016, G. Zaghloul [22] improved Hu's result, he proved T(x) << x 3/2 exp(-c(?)),where c > 0 is constant.In 1938, Hua [10] proved that almost all integers n ? 3( mod 24) and n(?)0(mod 5) can be expressed as a sum of three squares of primes. Later many results of the exceptional set of the sum of three squares of primes were proved (see [1], [13],[15], [17] etc.). In this paper, we will study problems similar to T(x), and our main results are as follows.Theorem 1. Let???(x)=? ?(p12+p22 +p23)?(p12+p22+p32.p13+p22+p32?x Then??,?(x) = O(x3/2 exp(-c(?))),where c > 0 is a constant.Theorem 2. When k ? 3 and s > min{2k-1, k2 + k-2} , let S(x)=? ?(m1k+…+msk)?(m1k+…+mks).m1k+…+msk?x Then S(x) =O(xs/k exp(-c'(?)),where c'> 0 is a constant.Theorem 3. When s > 3, let??(x)=??(m12+…+ms2).m12+…ms2?x Then??(x) = 2sCsIsx2/s+ O(x2/s log-Ax),where A > 0 is a constant, andTo prove these theorems, we follow the classical line of circle method. Because we do not know whether the Ramanujan Conjecture is true, in the present situation we only can use the mean estimate for Fourier coefficients of Maass cusp form. |
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