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1.
Gerald Kuba 《Acta Mathematica Hungarica》2002,95(1-2):115-124
For a positive real parameter t, real numbers , , and
, we consider sums
, where
is the rounding error function, i.e.\
. Generalizing and improving the main result of Part I of the paper we show that there exists an absolute constant
such that
for all
, and all
. Further, we give applications concerning the circle problem with linear, polynomial, and general weight. 相似文献
2.
Let r(n) count the number of representations of a positive integer n as a sum of two integer squares. We prove a truncated Voronoi-type formula for the twisted Mobius transform
where k and l are positive integers such that k and 4l are coprime, and give some applications (almost periodicity, limit distribution, an asymptotic mean-square formula, and O- and Ω-estimates for the error term).__________Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 10, pp. 1405 – 1418, October, 2004. 相似文献
3.
4.
Philippe Langlois 《BIT Numerical Mathematics》2001,41(3):515-539
A new automatic method to correct the first-order effect of floating point rounding errors on the result of a numerical algorithm is presented. A correcting term and a confidence threshold are computed using algorithmic differentiation, computation of elementary rounding error and running error analysis. Algorithms for which the accuracy of the result is not affected by higher order terms are identified. The correction is applied to the final result or to sensitive intermediate results to improve the accuracy of the computed result and/or the stability of the algorithm.This revised version was published online in October 2005 with corrections to the Cover Date. 相似文献
5.
We present a unified rounding error bound for polynomial evaluation. The bound presented here takes the same general form for the evaluation of a polynomial written in any polynomial basis when the evaluation algorithm can be expressed as a linear recurrence or a first-order linear matrix recurrence relation. Examples of these situations are: Horner's algorithm in the evaluation of power series, Clenshaw's and Forsythe's algorithms in the evaluation of orthogonal polynomial series, de-Casteljau's algorithm for Bernstein polynomial series, the modification of Clenshaw's algorithms in the evaluation of Szeg polynomial series, and so on. 相似文献
6.
This paper is to provide some new generalizations of the Pick Theorem. We first derive
a point-set version of the Pick Theorem for an arbitrary bounded lattice polyhedron. Then, we
use the idea of a weight function of [2] to obtain a weighted version. Other Pick type theorems
known to the author for the integral lattice
Z2 are reduced to some special cases of this generalization.
Finally, using an idea of Ehrhart [6] and the Pick Theorem, we give a direct proof of
the reciprocity law for Dedekind sums. The ideas and methods presented here may be pushed to
higher dimensions.AMS Subject Classification: 52C05, 11H06, 57N05, 57N15, 57N35. 相似文献
7.
In this paper, the authors use the analytic methods and the properties of
character sums mod p to study the computational problem of one kind of mean value
involving the classical Dedekind sums and two-term exponential sums, and give an exact
computational formula for it. 相似文献
8.
讨论三类整数列,这些数列的后项均是由前项与非整数乘积再取整后得到的,对应的取整函数分别为四舍五入取整函数、下取整函数、上取整函数.结果表明这三类整数列均为二阶线性递归数列. 相似文献
9.
Non-trivial estimates for fractional moments of smooth cubicWeyl sums are developed. Complemented by bounds for such sumsof use on both the major and minor arcs in a Hardy-Littlewooddissection, these estimates are applied to derive an upper boundfor the sth moment of the smooth cubic Weyl sum of the expectedorder of magnitude as soon as s> 7.691. Related argumentsdemonstrate that all large integers n are represented as thesum of eight cubes of natural numbers, all of whose prime divisorsare at most exp (c(log nlog log n)1/2}, for a suitable positivenumber c. This conclusion improves a previous result of G. Harcosin which nine cubes are required. 1991 Mathematics Subject Classification:11P05, 11L15, 11P55. 相似文献
10.
The arithmetic function rk–(n) counts the number of ways to write a natural number n as the difference of two kth powers (k 3 fixed). The investigation of the asymptotic behavior of the Dirichlet summatory function of rk–(n) leads in a natural way to a certain error term Pk–(t). In this article we establish a mean-square upper bound and an -estimate for Pk–(t). 相似文献
11.
Dedekind和的一个性质 总被引:4,自引:0,他引:4
Dedekind和的Knopp等式是与Hecke算子有关的一个算术性质,本文不借助eta-函数的概念,给予Knopp等式的一个简短的初等证明,同时把Knopp等式拓广到广义Dedekind和中。 相似文献
12.
本文利用三角和估计及其特征和的性质研究一类二项指数和与多项式特征和的混合均值的计算问题,并给出两个有趣的计算公式. 相似文献
13.
设整数q>2,c与q互素.对于1到q之间与q互素的任意整数a,在1到q之间存在唯一的整数b满足ab≡c mod q.对任意整数k≥2,定义M(q,k,c)为满足1≤ai≤q, (ai,q)=1,i=1,2,…,k,a1a2…ak≡c mod q且2 a1 a2 … ak的正整数组(a1,a2,…,ak)的数目,并设E(q,k,c)=M(q,k,c)-(φk-1(q))/2.本文的主要目的是利用Gauss和与原特征的性质,以及Dirichlet L-函数的均值定理,来研究E(q,k,c)与超级Kloosterman和K(h,k,q)的混合均值,并给出一个均值公式. 相似文献
14.
On a Problem Related to Sphere and Circle Packing 总被引:1,自引:0,他引:1
The paper proves that a set which contains spheres centeredat all points of a set of Hausdorff dimension greater than 1must have positive Lebesgue measure. It also proves the correspondingresult for circles, provided that the set of centers has Hausdorffdimension greater than 3/2. 相似文献
15.
本杠杆效应反映了股票收益率与其波动率变动之间的负相关关系,它一直是金融研究的核心问题.在高频时间序列数据中,传统的简单相关系数估计是不相合的,为此一些学者给出了新的杠杆效应刻画-积分杠杆效应,并给出该杠杆效应的估计量.众所周知,高频数据易受市场微观结构噪音的干扰,其中舍入误差是非常重要、实际中普遍存在的一类.高频数据被... 相似文献
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17.
The main purpose of this paper is using the analytic method and the properties of trigonometric sums and character sums to study the computational problem of one kind hybrid power mean involving two-term exponential sums and polynomial character sums. Then the authors give some interesting calculating formulae for them. 相似文献
18.
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. 相似文献
19.
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
20.
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 BombieriIwaniec 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) 140] 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 BombieriIwaniec method, which is to appearin Funct. Approx. Comment. Math. We turn the whole BombieriIwaniec 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). 相似文献