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 fi-nite field Fq , the digital quadratic exponential generator and quadratic exponential pseudorandom vector gen-erator. We...Based on the quadratic exponential method, this paper constructs two types of generators over fi-nite field Fq , the digital quadratic exponential generator and quadratic exponential pseudorandom vector gen-erator. 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 .If t is the least period of the sequence and t ≥ q1 /2 +2 ε,then the bound of the discrepancy is O (t ? 1/4 q1 /8+ εlog q)for any ε > 0.It shows that the sequence is asymptotically uniformly distributed.展开更多
In this paper, we prove the following estimateon exponential sums over primes: Let κ≥ 1, βκ = 1/2 + log κ/log 2, x ≥ 2 and α = a/q + λ subject to (a, q) = 1, 1 ≤ a ≤ q,and λ R. ThenΣx＜m≤2x∧(m)e(αmk)＜...In this paper, we prove the following estimateon exponential sums over primes: Let κ≥ 1, βκ = 1/2 + log κ/log 2, x ≥ 2 and α = a/q + λ subject to (a, q) = 1, 1 ≤ a ≤ q,and λ R. ThenΣx＜m≤2x∧(m)e(αmk)＜＜(d(q))βk(logx)c(x1/2√q(1+|λ|xk)+x4/5+x/√q(1+|λ|xk)).As an application, we prove that with at most O(N7/8+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
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
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
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 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 twisted T -adic exponential sum associated with xd + λx is studied. If λ = 0, then an explicitarithmetic polygon is proved to be the Newton polygon of the C-function of the twisted T-adic exponential sum.It give...The twisted T -adic exponential sum associated with xd + λx is studied. If λ = 0, then an explicitarithmetic polygon is proved to be the Newton polygon of the C-function of the twisted T-adic exponential sum.It gives the Newton polygons of the L-functions of twisted p-power order exponential sums.展开更多
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 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 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
In this paper, we establish a new estimate on exponential sums by using the Bombieri-type theorem and the modified Huxley-HoMey contour. We also generalize the famous Goldbach-Vinogradov theorem, via different argumen...In this paper, we establish a new estimate on exponential sums by using the Bombieri-type theorem and the modified Huxley-HoMey contour. We also generalize the famous Goldbach-Vinogradov theorem, via different argument from that of Vinogradov. In particular, our major arcs are quite large and these enlarged major arcs are treated by the estimate we have established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
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 fi-nite field Fq , the digital quadratic exponential generator and quadratic exponential pseudorandom vector gen-erator. 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 .If t is the least period of the sequence and t ≥ q1 /2 +2 ε,then the bound of the discrepancy is O (t ? 1/4 q1 /8+ εlog q)for any ε > 0.It shows that the sequence is asymptotically uniformly distributed.
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 estimateon exponential sums over primes: Let κ≥ 1, βκ = 1/2 + log κ/log 2, x ≥ 2 and α = a/q + λ subject to (a, q) = 1, 1 ≤ a ≤ q,and λ R. ThenΣx＜m≤2x∧(m)e(αmk)＜＜(d(q))βk(logx)c(x1/2√q(1+|λ|xk)+x4/5+x/√q(1+|λ|xk)).As an application, we prove that with at most O(N7/8+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.
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.
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supported by National Natural Science Foundation of China (Grant No.10671015)
文摘The twisted T -adic exponential sum associated with xd + λx is studied. If λ = 0, then an explicitarithmetic polygon is proved to be the Newton polygon of the C-function of the twisted T-adic exponential sum.It gives the Newton polygons of the L-functions of twisted p-power order exponential sums.
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 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.
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In this paper, we establish a new estimate on exponential sums by using the Bombieri-type theorem and the modified Huxley-HoMey contour. We also generalize the famous Goldbach-Vinogradov theorem, via different argument from that of Vinogradov. In particular, our major arcs are quite large and these enlarged major arcs are treated by the estimate we have established
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）.
