On Ternary Quadratic Forms Attached To Fourier Coefficients

In this paper, we study the mean value on ternary quadratic forms attached to Fourier coefficients. We establish its asymptotic formulas by using three-dimensional divisor problem and Perron’s formula, which extend the properties of exponential divisor function and have o significant meaning for the further research.Let f be a Maass cusp form with Laplace eigenvalue 1/4+v². Normalizing f with the first coefficient being 1, the Fourier expansion of f becomeswhere 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 [10]）. Applying the Theorem of K. Chandrasekharan and R. Narasimhan [1], we can deduce thatThe above bound will be used in our proof in the sequel.The ternary quadratic form m₁²+m₂²+m₃² also attracts the interest of many scholars and has been extensively studied. For instance, Vinogradov [13] and Chen[2] independently studied the well-known sphere problem and obtained the asymp-totic formulaSubsequently, the exponent 2/3 in the above error term was improved to 29/44 by Chamizo [3] and Iwaniec [11], and to 21/32 by Heath-Brown [9].Many authors studied a plenty of problems connected with this ternary quadratic form. Calderon and de Velasco [4] studied the average behavior of d（n） over values of ternary quadratic form and established the asymptotic formulaDefineRecently, Guo and Zhai [7] improved the average behavior of d（n） by showing thatand the error term was further improved by Zhao [14] to O（x2log7x）.In this paper, motivated by the above results, we consider a hybrid problem of the Fourier coefficients λ （n） of Maass cusp forms with the ternary quadratic form m₁²+m₂²+m₃² and quaternary quadratic form m₁²+m₂²+m₃²+m₄².We give an upper bound by presenting the following result.Theorem 1. LetThen we have S3（X） = 0(X^5/2log⁵X).Theorem 2. LetThen we have S4（X） = O（X³log⁶X）.To prove these result, we use the similar method in Zhao [14]. In fact, we use the Hardy-Littlewood-Kloosterman circle method, but the Voronoi type summation formula of λ（n） is also required. The difference is that here the asymptotic expan-sions of the J-Bessel function and K-Bessel function are more complicated, which bring some difficulties in computing the hybrid estimate of exponential sum and Fourier coefficients of Maass cusp form.