【作者】 徐哲峰；

【导师】 张文鹏；

【作者基本信息】 西北大学 ， 基础数学， 2007， 博士

【摘要】 本文主要研究了数论中一些和式的算术性质。这些和式包括不完整区间上的特征和、多元多项式特征和、hyper-Kloosterman和、带特征的指数和、Dedekind和以及它们的推广和式。此外，还研究了半区间上的D．H．Lehmer问题以及以及一些Smarandache型函数的算术性质。具体来说，本文主要包括以下几方面的结果：1．研究了带特征的完整三角和与两项指数和的高次均值，利用解析和初等的方法获得了它们四次均值的确切计算公式。2．讨论了不完整区间上特征和的高次均值的渐近性质。首先，分别对四分之一区间上偶原特征和与奇原特征和进行了研究，获得了它们的2κ次均值的渐近公式；第二，研究了八分之一区间上偶原特征和的2κ次均值，并获得了一个较强的渐近公式；第三，对四分之一区间上的非主特征和的四次均值进行了研究，获得了一个较强的渐近公式；第四，讨论了四分之一区间上原特征和的一次均值，同样也获得了一些渐近公式；最后，利用我们获得的四分之一区间上特征和的一个恒等式，推广并证明了著名的欧拉数猜想。3．讨论了多元多项式特征和的计算问题，对一类多元多项式特征和给出了确切的计算公式，并由此说明了Katz所获得的估计是最佳的。4．对经典的Dedekind和与它的类似和式Cochrane和定义了一种特殊形式的均值，并研究了它们的渐近性质。定义了高维Cochrane和，得到了高维Cochrane和的阶估计与平方均值渐近公式。5．研究了hyper-Kloosterman和的高次均值。在一定的条件限定下，给出了它的四次均值计算公式。6．研究了D．H．Lehmer问题误差项的一种均值，并获得了一些有趣而奇特的结果。把D．H．Lehmer问题限定在半区间上。研究了半区间上D．H．Lehmer问题的误差项，获得了误差项平方均值的一个较强的渐近公式。7．讨论了Smarandache函数的值分布性质和Smarandache幂函数的均值，得到了一些有趣的性质。更多还原

【Abstract】 The main purpose of this dissertation is to study the arithmetical properties of some summations in number theory. These summations are Character sums over incomplete intervals, Dirichlet characters of polynomials in several variables, Exponential sums with characters, hyper-Kloosterman sums, Dedekind sums and their generalized sums. Besides these, D. H. Lehmer problems and some Smarandache type functions were also studied. The main achievements contained in this dissertation are follows:1. Studied the hig.her power mean of complete trigonometric sums and two term exponential sums with Dirichlet characters, and obtained some exact calculation formulas for their fourth power mean by using analytical and elementary methods;2. Studied the asymptotic properties of higher power mean of Dirichlet character sums over incomplete intervals. At first, I studied the odd and even primitive character sums over quarter interval, respectively, and obtained some asymptotic formulas of their 2κth power mean; Secondly, I studied the 2κth power mean of even primitive character sums over one eighth interval and got a sharper asymptotic formula for it; Thirdly, I studied the fourth power mean of nonprincipal character sums over quarter interval and obtained a sharper asymptotic formula too; Fourthly, I studied the first power mean of primitive character sums over quarter interval and got some asymptotic formulas for it; At last, applying the identity for character sums over quarter interval which I obtained, generalized and proved the famous Euler numbers conjecture;3. Studied the evaluation problem of Dirichlet characters of polynomial in several variables and obtained some identities for a kind of Dirichlet characters of polynomial. This results explained the fact that Kats’s estimation is the best possible;4. Defined some special mean values for the classical Dedekind sums and its analogous sums which called Cochrane sums, and studied their asymptotic properties. Defined the high dimensional Cochrane sums and obtained its order estimation and asymptotic formula of its square mean value;5. Studied the higher power mean of hyper-Kloosterman sums and got an calculation formula for their fourth power mean under some restrictions;6. Studied a mean value of error term of D. H. Lehmer problem and obtained some interesting and strange results. Restricted D. H. Lehmer problem on a half interval. Studied the error term of the D. H. Lehmer problem over half interval and obtained a sharper asymptotic formula for its square mean value;7. Studied the value distribution of Smarandache function and the mean value of Smarandache power function, and obtained some interesting properties.更多还原