A generating function of higher-dimensional Apostol-Zagier sums and its reciprocity law

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.     

Some applications of smooth bilinear forms with Kloosterman sums

We revisit a recent bound of I. Shparlinski and T. Zhang on bilinear forms with Kloosterman sums, and prove an extension for correlation sums of Kloosterman sums against Fourier coefficients of modular forms. We use these bounds to improve on earlier results on sums of Kloosterman sums along the primes and on the error term of the fourth moment of Dirichlet L-functions.     

An identity involving Dedekind sums and generalized Kloosterman sums

The various properties of classical Dedekind sums S(h, q) have been investi-gated by many authors. For example, Yanni Liu and Wenpeng Zhang: A hybrid mean value related to the Dedekind sums and Kloosterman sums, Acta Mathematica Sinica, 27 (2011), 435–440 studied the hybrid mean value properties involving Dedekind sums and generalized Kloosterman sums K(m, n, r; q). The main purpose of this paper, is using the analytic methods and the properties of character sums, to study the computational problem of one kind of hybrid mean value involving Dedekind sums and generalized Kloosterman sums, and give an interesting identity.     

关于记录值序列部分和的渐近正态性

证明了 Burr分布和 Frechét分布的记录值序列的部分和是渐进对数正态的 ,从而解决了文[2 ]中的一个悬而未决的问题     

Dedekind sums in function fields

Okada (J Number Theory, 130:1750–1762, 2010) introduced Dedekind sums associated to a certain A-lattice, and established the reciprocity law. In this paper, we introduce Dedekind sums for arbitrary A-lattice and establish the reciprocity law for them. We next introduce higher dimensional Dedekind sums for any A-lattice. These Dedekind sums are analogues of Zagier’s higher dimensional Dedekind sums. We discuss the reciprocity law, rationality and characterization of these sums.     

Kloosterman sum identities and low-weight codewords in a cyclic code with two zeros

We apply relations of n-dimensional Kloosterman sums to exponential sums over finite fields to count the number of low-weight codewords in a cyclic code with two zeros. As a corollary we obtain a new proof for a result of Carlitz which relates one- and two-dimensional Kloosterman sums. In addition, we count some power sums of Kloosterman sums over certain subfields.     

A diophantine problem concerning cubes

This paper deals with the problem of finding n integers such that their pairwise sums are cubes. We obtain eight integers, expressed in parametric terms, such that all the six pairwise sums of four of these integers are cubes, 9 of the 10 pairwise sums of five of these integers are cubes, 12 pairwise sums of six of these integers are cubes, 15 pairwise sums of seven of these integers are cubes and 18 pairwise sums of all the eight integers are cubes. This leads to infinitely many examples of four positive integers such that all of their six pairwise sums are cubes. Further, for any arbitrary positive integer n, we obtain a set of 2(n+1) integers, in parametric terms, such that 5n+1 of the pairwise sums of these integers are cubes. With a choice of parameters, we can obtain examples with 5n+2 of the pairwise sums being cubes.     

A hybrid mean value of the Dedekind sums

The main purpose of this paper is to use the M. Toyoizumi’s important work, the properties of the Dedekind sums and the estimates for character sums to study a hybrid mean value of the Dedekind sums, and give a sharper asymptotic formula for it.     

Identities of incomplete Kloosterman sums

Identities between incomplete Kloosterman sums and incomplete hyper-Kloosterman sums are established.

     

A Kuznetsov Formula for Kloosterman Sums on GLn

We prove a Kuznetsov sum formula for Kloosterman sums on GL _n corresponding to the big Bruhat cell. Using this formula, a weighted sum of Kloosterman sums can be expressed in spectral decomposition. A non-trivial estimate of the spectral side might lead to a proof of cancellations in sums of Kloosterman sums on GL _n.     

On the density of Birkhoff sums for Anosov diffeomorphisms

Let f : M → M be an Anosov diffeomorphism on a nilmanifold. We consider Birkhoff sums for a Holder continuous observation along periodic orbits. We show that if there are two Birkhoff sums distributed at both sides of zero, then the set of Birkhoff sums of all the periodic points is dense in R.     

Some <Emphasis Type="Italic">q</Emphasis>-congruences for homogeneous and quasi-homogeneous multiple <Emphasis Type="Italic">q</Emphasis>-harmonic sums

We show some new Wolstenholme type q-congruences for some classes of multiple q-harmonic sums of arbitrary depth with strings of indices composed of ones, twos, and threes. Most of these results are q-extensions of the corresponding congruences for ordinary multiple harmonic sums obtained by the authors in a previous paper. We also establish duality congruences for multiple q-harmonic non-strict sums and a kind of duality for multiple q-harmonic strict sums. Finally, we pose a conjecture concerning two kinds of cyclic sums of multiple q-harmonic sums.     

On the 2<Emphasis Type="Italic">k</Emphasis>-th Power Mean of Inversion of <Emphasis Type="Italic">L</Emphasis>-functions with the Weight of the Gauss Sum

Abstract The main purpose of this paper is to use the estimate for character sums and the method of trigonometric sums to study the 2k-th power mean of the inversion of Dirichlet L-functions with the weight of the Gauss sums, and give a sharper asymptotic formula. This work is supported by the Doctorate Foundation of Xi’an Jiaotong University     

Infinite class of new sign ambiguities

In 1934, two kinds of multiplicative relations, the norm and the Davenport-Hasse relations, between Gauss sums, were known. In 1964, H. Hasse conjectured that the norm and the Davenport-Hasse relations were the only multiplicative relations connecting Gauss sums over Fp. However, in 1966, K. Yamamoto provided a simple counterexample disproving the conjecture. This counterexample was a new type of multiplicative relation, called a sign ambiguity, involving a ± sign not connected to elementary properties of Gauss sums. In this paper, we give an explicit product formula involving Gauss sums which generates an infinite class of new sign ambiguities, and we resolve the ambiguous sign by using Stickelberger?s theorem.     

Estimates on polynomial exponential sums

In this paper we establish new bounds on exponential sums of high degree for general composite moduli. The sums considered are either Gauss sums or ‘sparse’ and we rely on earlier work in the case of prime modulus.     

Estimates of Weyl Sums over Subsequences of Natural Number

In this paper we introduce the notion of pseudo -ergodicity to generalize Pustyl'nikov's estimates of Weyl sums to Weyl sums over subsequence of the natural numbers.     

Biproportional scaling of matrices and the iterative proportional fitting procedure

A short proof is given of the necessary and sufficient conditions for the convergence of the Iterative Proportional Fitting procedure. The input consists of a nonnegative matrix and of positive target marginals for row sums and for column sums. The output is a sequence of scaled matrices to approximate the biproportional fit, that is, the scaling of the input matrix by means of row and column divisors in order to fit row and column sums to target marginals. Generally it is shown that certain structural properties of a biproportional scaling do not depend on the particular sequence used to approximate it. Specifically, the sequence that emerges from the Iterative Proportional Fitting procedure is analyzed by means of the L ₁-error that measures how current row and column sums compare to their target marginals. As a new result a formula for the limiting L ₁-error is obtained. The formula is in terms of partial sums of the target marginals, and easily yields the other well-known convergence characterizations.     

Maximum and minimum weighted diagonal sums of certain non-negative matrices

This paper extends the notion of diagonal sums of a square matrix to “weighted diagonal sums”. Using simple probabilistic arguments, most of the results of Wang [5] concerning the maximum and minimum diagonal sums of doubly stochastic matrices are extended to maximum and minimum weighted diagonal sums of stochastic matrices (Sec. 3). Two stronger versions of one of Wang's conjectures are also proven (Theorems 4.1 and 5.1), of which the latter easily generalizes to the case of non-negative matrices (Theorem 5.2). The paper ends with a few open questions and counter-examples.     

Explicit formulae for sums of products of Bernoulli polynomials,including poly-Bernoulli polynomials

We study sums of products of Bernoulli polynomials, including poly-Bernoulli polynomials. As a main result, for any positive integer $m$ , explicit expressions of sums of $m$ products are given. This result extends that of the first author, as well as the famous Euler formula about sums of two products of Bernoulli numbers.