Mean Value on the Difference Between a Quadratic Residue and Its Inverse Modulo p

The main purpose of this paper is to use the generalized Bernoulli numbers, Gauss sums and the mean value theorems of Dirichlet L-functions between a quadratic residue and its inverse modulo p value formula. to study the asymptotic property of the difference （a prime）, and to give an interesting hybrid mean     

r次剩余及其关于模p的逆之差的混合均值

利用广义Bernoulli数,Gauss和与Dirichlet L-函数的均值来研究r次剩余及其关于模p的逆之差,并给出与广义Kloosterman和相关的一些混合均值公式.     

Gauss和与广义Kloosterman和的几个新的恒等式

主要应用Gauss和的性质和解析方法来研究Gauss和与广义Kloosterman和之间的关系,并且给出了几个有趣的恒等式.     

广义高阶Bernoulli数,Gauss和及广义Kloosterman和(英文)

利用广义高阶Bernoulli数的性质及Dirichlet L-函数的均值定理,研究了Gauss和及广义Kloosterman和与广义高阶Bernoulli数的均值性质,并给出两个有趣的渐近公式.     

三次高斯和与Kloosterman和的线性递推公式

应用三角和方法以及高斯和的若干性质,研究三次高斯和与Kloosterman和的一类高次混合均值的计算问题,本文给出该混合均值的一个有趣的线性递推公式.同时,还应用该递推公式,得到三次高斯和与Kloosterman和的高次混合均值的一系列较强的渐近公式.     

短区间的并集中整数及其m次幂的差的均值分布

本文研究了短区间的并集中整数及其m次幂的差的均值分布问题,给出了渐近公式.具体来说,设P是奇素数,1≤H≤p,实数δ满足0 δ≤1,整数m≥2.设I~((j))是(0,p)的互不相交的子区间,1≤j≤J,满足H/2≤|I~((j))|≤H,以及(y)_p表示y在模p下的非负最小剩余.定义I=∪_(j=1)~JI~((j)),并设X是模p的Dirichlet非主特征.证明了Σ x∈1 |x-(x~m)p|δp 1=1/p∫_0~([δp]) (Σ x∈1 x≤p-1-t 1+Σ x∈1 x≥t=1 1)dt+O(mJ~(1/2)P~(1/2)log~2 plog H),以及Σ x∈1 |x-(x~m)p|δp X(x)mJ~(1/2)P~(1/2)log~2 plog H.     

关于广义$k$次高斯和的均值

The main purpose of this paper is using residue system and character sums methods to investigate the mean value properties of general k-th Gauss sums,and two exact calculating formulas are given.     

广义k次Gauss和及其四次均值

本文研究了广义k次Gauss和的四次均值,给出了两个计算公式．     

关于模N的原根与它的逆的差的奇数幂及其推广

本文研究了模ｎ的原根ａ与它的逆ａ差的奇数幂分布性质，并给出了(?)的一个渐近公式．     

On the Residues of Binomial Coefficients and Their Products Modulo Prime Powers

In this paper, we show several arithmetic properties on the residues of binomial coefficients and their products modulo prime powers, e.g., for any distinct odd primes p and q. Meanwhile, we discuss the connections with the prime recognitions. Received November 5, 1998, Accepted December 7, 2000.     

三次Gauss和及二项指数和的混合均值

我们用三角和的性质研究一类三次Gauss和与两项指数和混合均值的计算问题,并给出一个精确的计算公式.     

On the difference between an integer and its inverse modulon (Ⅱ)

The main purpose of this paper is using the mean value theorem of Dirichlet L-functions to studythe asymptotic property of one class of number-theoretic functions, and give an interesting mean square valueformula.     

On the difference between an integer and its inverse modulo n(II)

The main purpose of this paper is using the mean value theorem of Dirichlet L-functions to study the asymptotic property of one class of number-theoretic functions, and give an interesting mean square value formula.     

On the difference between an integer and its inverse modulo <Emphasis Type="Italic">n</Emphasis>(II)

The main purpose of this paper is using the mean value theorem of Dirichlet L-functions to study the asymptotic property of one class of number-theoretic functions, and give an interesting mean square value formula.     

整值随机变量序列发生次数与条件概率和的等价性

在没有独立性、平稳性和相依性假设的条件下,利用分析方法讨论了整值随机变量序列取某个数值的次数与随机变量序列取该数值的条件概率和的等价性问题,建立并证明了若干强极限定理.     

Exponential Sums and the Riemann Zeta Function V

A Van der Corput exponential sum is S = exp (2 i f(m)) wherem has size M, the function f(x) has size T and = (log M) / log T < 1. There are different bounds for S in differentranges for . In the middle range where is near 1/over 2, . This bounds the exponent of growthof the Riemann zeta function on its critical line Re s = 1/over2. Van der Corput used an iteration which changed at each step.The Bombieri–Iwaniec method, whilst still based on meansquares, introduces number-theoretic ideas and problems. TheSecond Spacing Problem is to count the number of resonancesbetween short intervals of the sum, when two arcs of the graphof y = f'(x) coincide approximately after an automorphism ofthe integer lattice. In the previous paper in this series [Proc.London Math. Soc. (3) 66 (1993) 1–40] and the monographArea, lattice points, and exponential sums we saw that coincidenceimplies that there is an integer point close to some ‘resonancecurve’, one of a family of curves in some dual space,now calculated accurately in the paper ‘Resonance curvesin the Bombieri–Iwaniec method’, which is to appearin Funct. Approx. Comment. Math. We turn the whole Bombieri–Iwaniec method into an axiomatisedstep: an upper bound for the number of integer points closeto a plane curve gives a bound in the Second Spacing Problem,and a small improvement in the bound for S. Ends and cusps ofresonance curves are treated separately. Bounds for sums oftype S lead to bounds for integer points close to curves, andanother branching iteration. Luckily Swinnerton-Dyer's methodis stronger. We improve from 0.156140... in the previous paperand monograph to 0.156098.... In fact (32/205 + , 269/410 +) is an exponent pair for every > 0. 2000 Mathematics SubjectClassification 11L07 (primary), 11M06, 11P21, 11J54 (secondary).     

不完整区间上整数及其逆的差

设实数λ, δ 满足0 < λ, δ ≤ 1, n > 2 为整数. 本文研究了不完整区间[1,[λn]] 上整数与其模n逆的差. 定义S(n, λ, δ) = #{a : 1 ≤ a ≤ λn, (a, n) = 1, |a - a| < δn}, 本文给出了有关S(n, λ, δ) 的一些渐近公式.     

Explicit Evaluation of Euler and Related Sums

Ever since the time of Euler, the so-called Euler sums have been evaluated in many different ways. We give here a (presumably) new proof of the classical Euler sum. We show that several interesting analogues of the Euler sums can be evaluated by systematically analyzing some known summation formulas involving hypergeometric series. Many other identities related to the Euler sums are also presented. Research of the first author was supported by Korea Science and Engineering Foundation Grant R05-2003-10441-0. Research of the second author was supported by the Natural Sciences and Engineering Research Council of Canada Grant OGP0007353. 2000 Mathematics Subject Classification: Primary–11M06, 33B15, 33E20; Secondary–11M35, 11M41, 33C20     

广义Dedekind和与L-函数的一类恒等式

本文的主要目的是引入广义 Ｄｅｄｅｋｉｎｄ和,并由此推出 Ｄｉｒｉｃｈｌｅｔ Ｌ－函数的 一类恒等式．