In this paper,we use the analytic methods to study the mean value properties involving the classical Dedekind sums and two-term exponential sums,and give two sharper asymptotic formulae for it.
The main purpose of this article is to study the calculating problem of the sixth power mean of the two-term exponential sums,and give an interesting calculating formula for it.At the same time,the paper also provides...The main purpose of this article is to study the calculating problem of the sixth power mean of the two-term exponential sums,and give an interesting calculating formula for it.At the same time,the paper also provides a new and effective method for the study of the high order power mean of the exponential sums.展开更多
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 s...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.展开更多
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 exponentia...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 formuiu for it.展开更多
The main purpose of this paper is using the analytic method and the properties of the character sums to study the computational problem of one kind hybrid power mean involving the character sums of polynomials and the...The main purpose of this paper is using the analytic method and the properties of the character sums to study the computational problem of one kind hybrid power mean involving the character sums of polynomials and the two-term exponential sums,and give several interesting identities and asymptotic formulae for them.展开更多
We obtain upper bounds for mixed exponential sums of the type $S(\chi ,f,p^m ) = \sum\nolimits_{x = 1}^{p^n } {\chi (x)e} _{p^m } (ax^n + bx)$ where pm is a prime power with m? 2 and X is a multiplicative character (m...We obtain upper bounds for mixed exponential sums of the type $S(\chi ,f,p^m ) = \sum\nolimits_{x = 1}^{p^n } {\chi (x)e} _{p^m } (ax^n + bx)$ where pm is a prime power with m? 2 and X is a multiplicative character (mod pm). If X is primitive or p?(a, b) then we obtain |S(χ,f,p m)| ?2np 2/3 m . If X is of conductor p and p?( a, b) then we get the stronger bound |S(χ,f,p m)|?np m/2.展开更多
In this article, we analyze the lower bound of the divisibility of families of exponential sums for binomials over prime field. An upper bound is given for the lower bound, and, it is related to permutation polynomials.
Based on the quadratic exponential method, this paper constructs two types of generators over finite field Fq, the digital quadratic exponential generator and quadratic exponential pseudorandom vector generator. We in...Based on the quadratic exponential method, this paper constructs two types of generators over finite field Fq, the digital quadratic exponential generator and quadratic exponential pseudorandom vector generator. We investigate the distribution of the sequence generated by the generators, and present results about their one dimensional discrepancy. The proofs are based on the estimate of certain character sum over Fq. Ift is the least period of the sequence and t≥q^1/2＋2c, then the bound of the discrepancy is O（t^-1/4q^1/8＋τ logq） for any ε 〉 0. It shows that the sequence is asymptotically uniformly distributed.展开更多
In this paper, we study the solutions for variable-order time-fractional diffusion equations. A three-point combined compact difference (CCD) method is used to discretize the spatial variables to achieve sixth-order a...In this paper, we study the solutions for variable-order time-fractional diffusion equations. A three-point combined compact difference (CCD) method is used to discretize the spatial variables to achieve sixth-order accuracy, while the exponential-sum-approximation (ESA) is used to approximate the variable-order Caputo fractional derivative in the temporal direction, and a novel spatial sixth-order hybrid ESA-CCD method is implemented successfully. Finally, the accuracy of the proposed method is verified by numerical experiments.展开更多
In this paper, we prove the following estimate on exponential sums over primes: Let κ≥1,βκ=1/2+log κ/log2, x≥2 and α=a/q+ λ subject to (a, q) = 1, 1≤a≤q, and λ ∈ R. Then As an application, we prove that wi...In this paper, we prove the following estimate on exponential sums over primes: Let κ≥1,βκ=1/2+log κ/log2, x≥2 and α=a/q+ λ subject to (a, q) = 1, 1≤a≤q, and λ ∈ R. Then As an application, we prove that with at most O(N2/8+ε) exceptions, all positive integers up to N satisfying some necessary congruence conditions are the sum of three squares of primes. This result is as strong as what has previously been established under the generalized Riemann hypothesis.展开更多
指数和C(m,n,r,s;q)的高次幂均值计算与上界估计方面的研究与诸多数论问题联系密切,例如华林问题等。设p为奇素数,关注参数n=1,指数幂r=4,s=2条件下的一类二项指数和的四次幂均值计算问题。利用解析方法,借助Dirichlet特征的奇偶性、正...指数和C(m,n,r,s;q)的高次幂均值计算与上界估计方面的研究与诸多数论问题联系密切,例如华林问题等。设p为奇素数,关注参数n=1,指数幂r=4,s=2条件下的一类二项指数和的四次幂均值计算问题。利用解析方法,借助Dirichlet特征的奇偶性、正交性及特征和的性质,研究了形如C(m,1,4,2;p)的二项指数和的四次均值计算,给出了在素数p≡3 mod 4情况下上述二项指数和的一个精确的计算公式。同时,对于此类研究内容,该文也提出了一些有待解决的公开问题。展开更多
In this paper we establish one new estimate on exponential sums over primes in short intervals. As an application of this result, we sharpen Hua's result by proving that each sufficiently large integer N congruent...In this paper we establish one new estimate on exponential sums over primes in short intervals. As an application of this result, we sharpen Hua's result by proving that each sufficiently large integer N congruent to 5 modulo 24 can be written as N = p12+p22+p32+p42+p52, with |pj-(N/5)^(1/2)|≤U = N1/2-1/20+ε, where pj are primes. This result is as good as what one can obtain from the generalized Riemann hypothesis.展开更多
The twisted T-adic exponential sums associated to a polynomial in one variable are studied.An explicit arithmetic polygon in terms of the highest two exponents of the polynomial is proved to be a lower bound of the Ne...The twisted T-adic exponential sums associated to a polynomial in one variable are studied.An explicit arithmetic polygon in terms of the highest two exponents of the polynomial is proved to be a lower bound of the Newton polygon of the C-function of the twisted T-adic exponential sums.This bound gives lower bounds for the Newton polygon of the L-function of twisted p-power order exponential sums.展开更多
基金supported by National Natural Science Foundation of China (Grant No.11071194)
文摘In this paper,we use the analytic methods to study the mean value properties involving the classical Dedekind sums and two-term exponential sums,and give two sharper asymptotic formulae for it.
基金Supported by NSFC(Grant No.11771351)NSBRP(Grant No.2019JM-207)。
文摘The main purpose of this article is to study the calculating problem of the sixth power mean of the two-term exponential sums,and give an interesting calculating formula for it.At the same time,the paper also provides a new and effective method for the study of the high order power mean of the exponential sums.
基金the National Natural Science Foundation of China(No.11771351)the Natural Science Basic Research Plan in Shaanxi Province(No.2018JQ1093)。
文摘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.
基金supported by the National Natural Science Foundation of China(Nos.11371291,11471258)the Graduate Independent Innovation Fund of Northwest University(No.YZZ13071)
文摘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 formuiu for it.
基金Supported by NSF(Grant Nos.11771351 and 11826205)
文摘The main purpose of this paper is using the analytic method and the properties of the character sums to study the computational problem of one kind hybrid power mean involving the character sums of polynomials and the two-term exponential sums,and give several interesting identities and asymptotic formulae for them.
基金Tsinghua University and the NNSF of China for supporting his visit to China during the Fall of 2000This work was supported by the National Natural Science Foundation of China (Grant No. 19625102).
文摘We obtain upper bounds for mixed exponential sums of the type $S(\chi ,f,p^m ) = \sum\nolimits_{x = 1}^{p^n } {\chi (x)e} _{p^m } (ax^n + bx)$ where pm is a prime power with m? 2 and X is a multiplicative character (mod pm). If X is primitive or p?(a, b) then we obtain |S(χ,f,p m)| ?2np 2/3 m . If X is of conductor p and p?( a, b) then we get the stronger bound |S(χ,f,p m)|?np m/2.
文摘In this article, we analyze the lower bound of the divisibility of families of exponential sums for binomials over prime field. An upper bound is given for the lower bound, and, it is related to permutation polynomials.
基金Supported by the Special Fund of National Excellent Doctoral Dissertation (Grant 200060) and the National Natural Science Foundation of China (No.60373092).
文摘Based on the quadratic exponential method, this paper constructs two types of generators over finite field Fq, the digital quadratic exponential generator and quadratic exponential pseudorandom vector generator. We investigate the distribution of the sequence generated by the generators, and present results about their one dimensional discrepancy. The proofs are based on the estimate of certain character sum over Fq. Ift is the least period of the sequence and t≥q^1/2＋2c, then the bound of the discrepancy is O（t^-1/4q^1/8＋τ logq） for any ε 〉 0. It shows that the sequence is asymptotically uniformly distributed.
文摘In this paper, we study the solutions for variable-order time-fractional diffusion equations. A three-point combined compact difference (CCD) method is used to discretize the spatial variables to achieve sixth-order accuracy, while the exponential-sum-approximation (ESA) is used to approximate the variable-order Caputo fractional derivative in the temporal direction, and a novel spatial sixth-order hybrid ESA-CCD method is implemented successfully. Finally, the accuracy of the proposed method is verified by numerical experiments.
基金The author is supported by Post-Doctoral Fellowsbip of The University of Hong Kong.
文摘In this paper, we prove the following estimate on exponential sums over primes: Let κ≥1,βκ=1/2+log κ/log2, x≥2 and α=a/q+ λ subject to (a, q) = 1, 1≤a≤q, and λ ∈ R. Then As an application, we prove that with at most O(N2/8+ε) exceptions, all positive integers up to N satisfying some necessary congruence conditions are the sum of three squares of primes. This result is as strong as what has previously been established under the generalized Riemann hypothesis.
文摘指数和C(m,n,r,s;q)的高次幂均值计算与上界估计方面的研究与诸多数论问题联系密切,例如华林问题等。设p为奇素数,关注参数n=1,指数幂r=4,s=2条件下的一类二项指数和的四次幂均值计算问题。利用解析方法,借助Dirichlet特征的奇偶性、正交性及特征和的性质,研究了形如C(m,1,4,2;p)的二项指数和的四次均值计算,给出了在素数p≡3 mod 4情况下上述二项指数和的一个精确的计算公式。同时,对于此类研究内容,该文也提出了一些有待解决的公开问题。
基金supported by the National Natural Science Foundation of China(Grant Nos.10125101&10531060)a Major Grant Program in Science and Technology by the Ministry of EducationTianyuan Mathematics Foundation(Grant No.10526028).
文摘In this paper we establish one new estimate on exponential sums over primes in short intervals. As an application of this result, we sharpen Hua's result by proving that each sufficiently large integer N congruent to 5 modulo 24 can be written as N = p12+p22+p32+p42+p52, with |pj-(N/5)^(1/2)|≤U = N1/2-1/20+ε, where pj are primes. This result is as good as what one can obtain from the generalized Riemann hypothesis.
基金supported by National Natural Science Foundation of China (Grant No.10671015)
文摘The twisted T-adic exponential sums associated to a polynomial in one variable are studied.An explicit arithmetic polygon in terms of the highest two exponents of the polynomial is proved to be a lower bound of the Newton polygon of the C-function of the twisted T-adic exponential sums.This bound gives lower bounds for the Newton polygon of the L-function of twisted p-power order exponential sums.
