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.
We study the exponential sums involving l:burmr coeffcients ot Maass forms and exponential functions of the form e(anZ), where 0 ≠ α∈R and 0 〈 β 〈 1. An asymptotic formula is proved for the nonlinear exponent...We study the exponential sums involving l:burmr coeffcients ot Maass forms and exponential functions of the form e(anZ), where 0 ≠ α∈R and 0 〈 β 〈 1. An asymptotic formula is proved for the nonlinear exponential sum ∑x〈n≤2x λg(n)e(αnβ), when β = 1/2 and |α| is close to 2√ q C Z+, where Ag(n) is the normalized n-th Fourier coefficient of a Maass cusp form for SL2 (Z). The similar natures of the divisor function 7(n) and the representation function r(n) in the circle problem in nonlinear exponential sums of the above type are also studied.展开更多
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
This is an expository paper on algebraic aspects of exponential sums over finite fields.This is a new direction.Various examples,results and open problems are presented along the way,with particular emphasis on Gauss ...This is an expository paper on algebraic aspects of exponential sums over finite fields.This is a new direction.Various examples,results and open problems are presented along the way,with particular emphasis on Gauss periods,Kloosterman sums and one variable exponential sums.One main tool is the applications of various p-adic methods.For this reason,the author has also included a brief exposition of certain p-adic estimates of exponential sums.The material is based on the lectures given at the 2020 online number theory summer school held at Xiamen University.Notes were taken by Shaoshi Chen and Ruichen Xu.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
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
The generic Newton polygon of L-functions associated with the exponential sums of poly- nomials of degree 3 in two variables is studied by Dwork's analytic methods. Wan's conjecture is shown to be true for this case.
Let p be an odd prime and let δ be a fixed real number with 0<δ<2.For an integer α with 0<α<p,denote by ā the unique integer between 0 and p satisfying aā≡1(mod p).Further, let{x}denote the fraction...Let p be an odd prime and let δ be a fixed real number with 0<δ<2.For an integer α with 0<α<p,denote by ā the unique integer between 0 and p satisfying aā≡1(mod p).Further, let{x}denote the fractional part of x.We derive an asymptotic formula for the number of pairs of integers(a,b)with 1≤a≤p-1,1≤b≤p-1,|{a^k/p}+{b^k/p}-{(?)~l/p}-{(?)~l/p}|<δ.展开更多
Let Af(n) be the coefficient of the logarithmic derivative for the Hecke L-function. In this paper we study the cancellation of the function Ay(n) twisted with an additive character e(α√n), α 〉0, i.e. Ef(x...Let Af(n) be the coefficient of the logarithmic derivative for the Hecke L-function. In this paper we study the cancellation of the function Ay(n) twisted with an additive character e(α√n), α 〉0, i.e. Ef(x) = Σx〈n〈2x Af(n)e(α√n).展开更多
Let F_(q) be a finite field and F_(q)^(s) be an extension of F_(q).Let f(x)∈F_(q)[x]be a polynomial of degree n with g c d(n,q)=1.We present a recursive formula for evaluating the exponential sum∑c∈F_(q)^(s)χ^((s)...Let F_(q) be a finite field and F_(q)^(s) be an extension of F_(q).Let f(x)∈F_(q)[x]be a polynomial of degree n with g c d(n,q)=1.We present a recursive formula for evaluating the exponential sum∑c∈F_(q)^(s)χ^((s))(f(x)).Let a and b be two elements in F_(q) with a a≠0,u be a positive integer.We obtain an estimate for the exponential sum∑c∈F^(∗)_(q)^(s)χ^((s))(ac^(u)+bc^(−1)),whereχ^((s))is the lifting of an additive characterχof F_(q).Some properties of the sequences constructed from these exponential sums are provided too.展开更多
Based on the theory of exponential sums an d quadratic forms over finite field, the crosscorrelation function values betwee n two maximal linear recursive sequences are determined under some conditions.
Two new families of finite binary sequences are constructed using multiplicative inverse. The sequences are shown to have strong pseudorandom properties by using some estimates of certain exponential sums over finite ...Two new families of finite binary sequences are constructed using multiplicative inverse. The sequences are shown to have strong pseudorandom properties by using some estimates of certain exponential sums over finite fields. The constructions can be implemented fast since multiplicative inverse over finite fields can be computed in polynomial time.展开更多
The Hodge bound for the Newton polygon of L-functions of T-adic exponential sums associated to a Laurent polynomial is established.We improve the lower bound and study the properties of this new bound.We also study wh...The Hodge bound for the Newton polygon of L-functions of T-adic exponential sums associated to a Laurent polynomial is established.We improve the lower bound and study the properties of this new bound.We also study when this new bound is reached with large p arbitrarily,and hence the generic Newton polygon is determined.展开更多
Let g be a holomorphic or Maass Hecke newform of level D and nebentypus XD, and let ag(n) be its n-th Fourier coefficient. We consider the sum S1=∑ X〈n≤2x ag(n)e(an β) and prove that S1 has an asymptotic for...Let g be a holomorphic or Maass Hecke newform of level D and nebentypus XD, and let ag(n) be its n-th Fourier coefficient. We consider the sum S1=∑ X〈n≤2x ag(n)e(an β) and prove that S1 has an asymptotic formula when β = 1/2 and α is close to ±2 √q/D for positive integer q ≤ X/4and X sufficiently large. And when 0 〈β 〈 1 and α, β fail to meet the above condition, we obtain upper bounds of S1. We also consider the sum S2 = ∑n〉0 ag(n)e(an β) Ф(n/X) with Ф(x) ∈ C c ∞(0,+∞) and prove that S2 has better upper bounds than S1 at some special α and β.展开更多
Let Pi, 1≤i≤5, be prime numbers. It is proved that every integer N that satisfies N=5 (mod 24) can be written as N=p1^2+p2^2+P3^2+p4^2 +p5^2, where │√N5-Pi│≤N^1/2-19/850+∈.
In this paper, we prove that each sufficiently large integer N ≠1(mod 3) can be written as N=p+p1^2+p2^2+p3^2+p4^2, with|p-N/5|≤U,|pj-√N/5|≤U,j=1,2,3,4,where U=N^2/20+c and p,pj are primes.
We prove that each sufficiently large odd integer N can be written as sum of the form N = p1^3 +p2^3 +... +p9^3 with [pj - (N/9)^1/31 ≤ N^(1/3)-θ, where pj, j = 1,2,...,9, are primes and θ = (1/51) -ε.
In this paper,we propose a new numerical scheme for the coupled Stokes-Darcy model with the Beavers-Joseph-Saffman interface condition.We use the weak Galerkin method to discretize the Stokes equation and the mixed fi...In this paper,we propose a new numerical scheme for the coupled Stokes-Darcy model with the Beavers-Joseph-Saffman interface condition.We use the weak Galerkin method to discretize the Stokes equation and the mixed finite element method to discretize the Darcy equation.A discrete inf-sup condition is proved and the optimal error estimates are also derived.Numerical experiments validate the theoretical analysis.展开更多
This paper shows a connection between exponential sums and character sums. In particular, we introduce a character sum that is an analog of the classical Kloosterman sums and establish the analogous Weil-Estermann...This paper shows a connection between exponential sums and character sums. In particular, we introduce a character sum that is an analog of the classical Kloosterman sums and establish the analogous Weil-Estermann's upper bound for it. The paper also analyzes a generalized Hardy-Littlewood example for character sums, which shows that the upper bounds given here are the best possible. The analysis makes use of local bounds for the exponential sums and character sums. The basic theorems have been previously established.展开更多
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 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).
基金Acknowledgements This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 11101239, 10971119), the Program for Changjiang Scholars and Innovative Research Team in University (IRT1264), and the Independent Innovation Foundation of Shandong University (Grant No. 2012ZRYQ005).
文摘We study the exponential sums involving l:burmr coeffcients ot Maass forms and exponential functions of the form e(anZ), where 0 ≠ α∈R and 0 〈 β 〈 1. An asymptotic formula is proved for the nonlinear exponential sum ∑x〈n≤2x λg(n)e(αnβ), when β = 1/2 and |α| is close to 2√ q C Z+, where Ag(n) is the normalized n-th Fourier coefficient of a Maass cusp form for SL2 (Z). The similar natures of the divisor function 7(n) and the representation function r(n) in the circle problem in nonlinear exponential sums of the above type are also studied.
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partially supported by the National Natural Science of Foundation under Grant No.1900929。
文摘This is an expository paper on algebraic aspects of exponential sums over finite fields.This is a new direction.Various examples,results and open problems are presented along the way,with particular emphasis on Gauss periods,Kloosterman sums and one variable exponential sums.One main tool is the applications of various p-adic methods.For this reason,the author has also included a brief exposition of certain p-adic estimates of exponential sums.The material is based on the lectures given at the 2020 online number theory summer school held at Xiamen University.Notes were taken by Shaoshi Chen and Ruichen Xu.
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 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Supported by National Natural Science Foundation of China (Grant No. 10671015)
文摘The generic Newton polygon of L-functions associated with the exponential sums of poly- nomials of degree 3 in two variables is studied by Dwork's analytic methods. Wan's conjecture is shown to be true for this case.
文摘Let p be an odd prime and let δ be a fixed real number with 0<δ<2.For an integer α with 0<α<p,denote by ā the unique integer between 0 and p satisfying aā≡1(mod p).Further, let{x}denote the fractional part of x.We derive an asymptotic formula for the number of pairs of integers(a,b)with 1≤a≤p-1,1≤b≤p-1,|{a^k/p}+{b^k/p}-{(?)~l/p}-{(?)~l/p}|<δ.
基金This work is supported by the National Natural Science Foundation of China (Grant No. 10701048)
文摘Let Af(n) be the coefficient of the logarithmic derivative for the Hecke L-function. In this paper we study the cancellation of the function Ay(n) twisted with an additive character e(α√n), α 〉0, i.e. Ef(x) = Σx〈n〈2x Af(n)e(α√n).
文摘Let F_(q) be a finite field and F_(q)^(s) be an extension of F_(q).Let f(x)∈F_(q)[x]be a polynomial of degree n with g c d(n,q)=1.We present a recursive formula for evaluating the exponential sum∑c∈F_(q)^(s)χ^((s))(f(x)).Let a and b be two elements in F_(q) with a a≠0,u be a positive integer.We obtain an estimate for the exponential sum∑c∈F^(∗)_(q)^(s)χ^((s))(ac^(u)+bc^(−1)),whereχ^((s))is the lifting of an additive characterχof F_(q).Some properties of the sequences constructed from these exponential sums are provided too.
文摘Based on the theory of exponential sums an d quadratic forms over finite field, the crosscorrelation function values betwee n two maximal linear recursive sequences are determined under some conditions.
基金Supported by the Open Funds of Key Lab of Fujian Province University Network Security and Cryptology (07B005)the Funds of the Education Department of Fujian Province (JA07164)the Natural Science Foundation of Fujian Province of China (2007F3086)
文摘Two new families of finite binary sequences are constructed using multiplicative inverse. The sequences are shown to have strong pseudorandom properties by using some estimates of certain exponential sums over finite fields. The constructions can be implemented fast since multiplicative inverse over finite fields can be computed in polynomial time.
文摘The Hodge bound for the Newton polygon of L-functions of T-adic exponential sums associated to a Laurent polynomial is established.We improve the lower bound and study the properties of this new bound.We also study when this new bound is reached with large p arbitrarily,and hence the generic Newton polygon is determined.
基金This work was supported in part by the Natural Science Foundation of Shandong Province (No. ZR2015AM016).
文摘Let g be a holomorphic or Maass Hecke newform of level D and nebentypus XD, and let ag(n) be its n-th Fourier coefficient. We consider the sum S1=∑ X〈n≤2x ag(n)e(an β) and prove that S1 has an asymptotic formula when β = 1/2 and α is close to ±2 √q/D for positive integer q ≤ X/4and X sufficiently large. And when 0 〈β 〈 1 and α, β fail to meet the above condition, we obtain upper bounds of S1. We also consider the sum S2 = ∑n〉0 ag(n)e(an β) Ф(n/X) with Ф(x) ∈ C c ∞(0,+∞) and prove that S2 has better upper bounds than S1 at some special α and β.
文摘Let Pi, 1≤i≤5, be prime numbers. It is proved that every integer N that satisfies N=5 (mod 24) can be written as N=p1^2+p2^2+P3^2+p4^2 +p5^2, where │√N5-Pi│≤N^1/2-19/850+∈.
基金the National Natural Science Foundation of China (Grant No.10701048)
文摘In this paper, we prove that each sufficiently large integer N ≠1(mod 3) can be written as N=p+p1^2+p2^2+p3^2+p4^2, with|p-N/5|≤U,|pj-√N/5|≤U,j=1,2,3,4,where U=N^2/20+c and p,pj are primes.
文摘We prove that each sufficiently large odd integer N can be written as sum of the form N = p1^3 +p2^3 +... +p9^3 with [pj - (N/9)^1/31 ≤ N^(1/3)-θ, where pj, j = 1,2,...,9, are primes and θ = (1/51) -ε.
基金National Natural Science Foundation of China(Grant No.11801318)Natural Science Foundation of Shandong Province(Grant No.ZR2018QA004)+2 种基金supported by National Natural Science Foundation of China(Grant Nos.11771252 and 11531008)the Ministry of Education of China(Grant No.IRT16R43)Taishan Scholars Project。
文摘In this paper,we propose a new numerical scheme for the coupled Stokes-Darcy model with the Beavers-Joseph-Saffman interface condition.We use the weak Galerkin method to discretize the Stokes equation and the mixed finite element method to discretize the Darcy equation.A discrete inf-sup condition is proved and the optimal error estimates are also derived.Numerical experiments validate the theoretical analysis.
基金Supported by the National Natural Science Foundationof China(No.196 2 5 10 2 ) and partially by the National"973"Project of China
文摘This paper shows a connection between exponential sums and character sums. In particular, we introduce a character sum that is an analog of the classical Kloosterman sums and establish the analogous Weil-Estermann's upper bound for it. The paper also analyzes a generalized Hardy-Littlewood example for character sums, which shows that the upper bounds given here are the best possible. The analysis makes use of local bounds for the exponential sums and character sums. The basic theorems have been previously established.