Maass尖形式傅里叶系数在整变量三元二次型中的均值分布

发布时间：2018-04-09 21:00

本文选题：傅里叶系数　切入点：二次型　出处：《山东师范大学》2017年硕士论文

【摘要】：本文主要研究了关于Maass尖形式傅里叶系数在整变量三元二次型中的均值分布。论文主要运用经典圆法与指数和来估计该均值问题的上界,丰富了关于傅里叶系数性质的结果,并对其进一步研究有重要的理论意义。令f是具有拉普拉斯特征值1/4+v~2的Maass尖形式。首项系数为1的标准化的f的傅里叶展开式为其中Ks(y)s=1/2+it的K-Bessel函数。关于f的L函数定义为当Re＞ 1时,上述级数是一致收敛的,见[10]。利用K. Chandrasekharan和R.Narasimhan[1]中的定理,我们可以得到在后面的证明中,我们需要利用这个上界。三元二次型m_1~2+m_2~2+m_3~2吸引了很多学者的关注并且得到了广泛的研究。比如,Vinogradov [13]和Chen [2]独立地研究了著名的球体问题并且得到了渐近公式后来,Chamizo [3]和Iwaniec[11]把余项中的指数2/3改进为29/44,Heath-Brown [9]又改进为 21/32。许多学者研究了大量关于三元二次型的问题。其中Calderon和de Velasco[4]研究了关于d(n)的三元二次型的均值问题,并建立了渐近公式令定义最近,Guo和Zhai [7]改进了关于d(n)的三元二次型的均值估计,得到并且Zhao [14]把其中的余项进一步改进为O(x~2log~7 x)。基于上述结论,在本文中,我们研究关于Maass尖形式傅里叶系数在整变量三元二次型m_1~2+m_2~2+m_3~2及四元二次型m_1~2+m_2~2+m_3~2+m_4~2中的均值分布。我们通过下面的结论给出它们的上界。定理1.令我们有S_3(X) =O(X~(5/2)log~5X).定理2.令我们有S_4(X) = O(X~3log~6X).为了证明定理结论,我们利用与Zhao[14]相似的方法。事实上,我们要使用Hardy-Littlewood-Kloosterman圆法,同时也需要利用关于λ(n)的Voronoi求和公式。不同的是,这里的J-Bessel函数和K-Bessel函数的渐近公式更复杂,这会使得Maass尖形式的傅里叶系数与指数和的混合估计更加困难。
[Abstract]:In this paper, we study the mean distribution of Fourier coefficients in the form of Maass cusp in ternary quadratic forms of integral variables.This paper mainly uses the classical circular method and the exponential sum to estimate the upper bound of the mean value problem, which enriches the results on the properties of Fourier coefficients and has important theoretical significance for its further study.Let f be a Maass tip form with a Laplace eigenvalue of 1 / 4 VX 2.The first normalized Fourier expansion of f with a coefficient of 1 is the K-Bessel function of Ks(y)s=1/2 it.The L function of f is defined as uniformly convergent when re > 1, see [10].By using the theorems in K. Chandrasekharan and R.Narasimhan [1], we can get that we need to use this upper bound in the following proof.The ternary quadratic type mStuff 2mS 2mS-1 / 2mSZ has attracted the attention of many scholars and has been widely studied.For example, Vinogradov [13] and Chen [2] have studied the famous sphere problem independently and obtained the asymptotic formula. Later, Iwaniec [3] and Iwaniec [11] improved the index 2 / 3 in the remainder to 29 / 4 Heath-Brown [9] and then 21 / 32.Many scholars have studied a lot of problems about ternary quadratic forms.In this paper, Calderon and de Velasco [4] study the mean value problem of ternary quadratic forms on dnn, and establish an asymptotic formula such that the definitions of Guo and Zhai [7] improve the mean estimation of ternary quadratic forms on dnn.And Zhao [14] further improved the rest of the term to O(x~2log~7 XG.We give their upper bounds by the following conclusions.Theorem 1.So we have 5 / 2 / 5 / 5 / 5 / 5 / 5 / 5.Theorem 2.So that we have S / S / 4 / X) = O / X / 3log/ 6X / _ _ _.In order to prove the theorem, we use a method similar to that of Zhao [14].In fact, we need to use the Hardy-Littlewood-Kloosterman circle method as well as the Voronoi summation formula about 位 n.The difference is that the asymptotic formulas of J-Bessel function and K-Bessel function are more complex, which makes the mixed estimation of Fourier coefficients and exponential sums in the form of Maass sharp more difficult.
【学位授予单位】：山东师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O174.2

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