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1.
In this paper, the generalized Cochrane sums and Cochrane-Hardy sums are defined. The arithmetic properties of the generalized Cochrane sums are studied, and the Cochrane-Hardy sums are expressed in terms of the generalized Cochrane sums. Analogues of Subrahmanyam's identity and Knopp's theorem are given and proved. Finally, the hybrid mean value of generalized Cochrane sums, Cochrane-Hardy sums and Kloosterman sums is studied, and a few asymptotic formulae are obtained.  相似文献   

2.
An upper bound estimate of high-dimensional Cochrane sums is given in this note using Weinstein's version of Deligne's estimate for the hyper-Kloosterman sum and a mean value theorem of Dirichlet L-functions.  相似文献   

3.
 The main purpose of this paper is using the estimate for classical Kloostermann sums and E. Bombieri–A. I. Vinogradov’s important work to study the first power mean of the inversion of Dirichlet L-functions with the weight of general Kloostermann sums, and give an interesting asymptotic formula. Received 30 March 2001; in revised form 12 November 2001  相似文献   

4.
While most proofs of the Weil bound on one-variable Kloosterman sums over finite fields are carried out in all characteristics, the original proof of this bound, by Weil, assumes the characteristic is odd. We show how to make Weil's argument work in even characteristic, for both ordinary Kloosterman sums and sums twisted by a multiplicative character.  相似文献   

5.
We introduce higher-dimensional Dedekind sums with a complex parameter z, generalizing Zagier's higher-dimensional Dedekind sums. The sums tend to Zagier's higher-dimensional Dedekind sums as z→∞. We show that the sums turn out to be generating functions of higher-dimensional Apostol-Zagier sums which are defined to be hybrids of Apostol's sums and Zagier's sums. We prove reciprocity law for the sums. The new reciprocity law includes reciprocity formulas for both Apostol and Zagier's sums as its special case. Furthermore, as its application we obtain relations between special values of Hurwitz zeta function and Bernoulli numbers, as well as new trigonometric identities.  相似文献   

6.
We find an expression for a sum which can be viewed as a generalization of power moments of Kloosterman sums studied by Kloosterman and Salié. Received: 24 March 2006  相似文献   

7.
In this paper, we use the properties of Gauss sums, primitive characters and the mean value theorems of Dirichlet L-functions to study the hybrid mean value of Cochrane sums and general Kloosterman sums, and give two sharp asymptotic formulae.  相似文献   

8.
We give nontrivial bounds in various ranges for exponential sums of the form
  相似文献   

9.
In this paper we utilize the estimation of number of solutions of congruence to obtain the upper bound of incomplete Kloosterman sums, which improves the Shparlinski's result and removes the parameter r.  相似文献   

10.
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.  相似文献   

11.
薛西锋 《数学学报》2007,50(6):1411-141
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.  相似文献   

12.
Gauss和与广义Kloosterman和的几个新的恒等式   总被引:5,自引:1,他引:4  
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.  相似文献   

13.
We consider Weyls exponential sums for very small values of the variable. In this case we can give an asymptotic transformation formula. Weyls exponential sums will be usually estimated by means of Weyls or Vinogradovs method. Here we use van der Corputs method and obtain sufficiently good results in the present case.Received: 14 January 2002  相似文献   

14.
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.  相似文献   

15.
Let χ be a Dirichlet character and L(s,χ) be its L-function. Using weighted averages of Gauss and Ramanujan sums, we find exact formulas involving Jordan?s and Euler?s totient function for the mean square average of L(1,χ) when χ ranges over all odd characters modulo k and L(2,χ) when χ ranges over all even characters modulo k. In principle, using our method, it is always possible to find the mean square average of L(r,χ) if χ and r?1 have the same parity and χ ranges over all odd (or even) characters modulo k, though the required calculations become formidable when r?3. Consequently, we see that for almost all odd characters modulo k, |L(1,χ)|<Φ(k), where Φ(x) is any function monotonically tending to infinity.  相似文献   

16.
We investigate certain character sums and prove some discrepancy-type inequalities for incomplete sums.   相似文献   

17.
We give upper bounds for the absolute value of exponential sums in several variables attached to certain polynomials with coefficients in a finite field. This bounds are given in terms of invariants of the singularities of the projective hypersurface defined by its highest degree form. For exponential sums attached to the reduction modulo a power of a large prime of a polynomial f with integer coefficients and veryfying a certain condition on the singularities of its highest degree form, we give a bound in terms of the dimension of the Jacobian quotient . Received: 3 November 1997  相似文献   

18.
We consider incomplete exponential sums in several variables of the form
  相似文献   

19.
Let q?2 be an integer, χ be any non-principal character mod q, and H=H(q)?q. In this paper the authors prove some estimates for character sums of the form
  相似文献   

20.
关于广义Dedekind和的2k次均值   总被引:4,自引:0,他引:4  
谢敏  张文鹏 《数学学报》2001,44(1):85-94
本文利用 Dirichlet L-函数的均值定理,研究广义 Dedekind和的渐近性质,并给出一个有趣的渐近公式.  相似文献   

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