The main purpose of this paper is using estimates for trigonometric sums and properties of congruence to study the computation of one kind of fourth power mean of a generalized three-term exponential sum, and give an ...The main purpose of this paper is using estimates for trigonometric sums and properties of congruence to study the computation of one kind of fourth power mean of a generalized three-term exponential sum, and give an interesting identity for it.展开更多
The computational problem of fourth power mean of generalized three-term exponential sums is studied by using the trigonometric identity and the properties of the reduced residue system. Some explicit formulas for the...The computational problem of fourth power mean of generalized three-term exponential sums is studied by using the trigonometric identity and the properties of the reduced residue system. Some explicit formulas for the fourth power mean of generalized three-term exponential sums under different conditions are given.展开更多
Let G be a strongly connected directed weighted graph with vertex set{v_(1),v_(2),...,v_(n)},in which each edge e is assigned with an arbitrary nonzero weight w(e).For any two vertices v_i,v_(j )of G,the distance d_(i...Let G be a strongly connected directed weighted graph with vertex set{v_(1),v_(2),...,v_(n)},in which each edge e is assigned with an arbitrary nonzero weight w(e).For any two vertices v_i,v_(j )of G,the distance d_(ij )from v_(i)to v_(j)is defined as dij=P∈P(v_(i),v_(j))min∑w(e) where P(v_(i),v_(j)) denotes the set consisting of all the directed paths from v_(i)to v_(j)in G.Given a nonzero indeterminant q,following the definitions from Yan and Yeh (Adv.Appl.Math.,2007),and Bapat et al.(Linear Algebra Appl.,2006),one can define the exponential distance matrix of G as F^(q)_(G)=(q^(dij))_(n×n),and define the q-distance matrix of G as D_(G)^(q)=(d_(ij)^(q))_(n×n)with d_(ij)^(q)={1-q^(dij)/1-q,if q≠1,dij,if q=1,extending the original definitions only for the undirected unweighted connected graphs.One of the remarkable results about the distance matrices of graphs is due to the Graham-HoffmanHosoya theorem (J.Graph Theory,1977).In this paper,we present some Graham-HoffmanHosoya type theorems for the exponential distance matrix F_(G)^(q)and q-distance matrix D_(G)^(q),extending all the known Graham-Hoffman-Hosoya type theorems.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.
In this paper,an exponential inequality for weighted sums of identically distributed NOD （negatively orthant dependent） random variables is established,by which we obtain the almost sure convergence rate of which re...In this paper,an exponential inequality for weighted sums of identically distributed NOD （negatively orthant dependent） random variables is established,by which we obtain the almost sure convergence rate of which reaches the available one for independent random variables in terms of Berstein type inequality. As application,we obtain the relevant exponential inequality for Priestley-Chao estimator of nonparametric regression estimate under NOD samples,from which the strong consistency rate is also obtained.展开更多
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
Suppose thatλ_(1),λ_(2),λ_(3),λ_(4),λ_(5)are nonzero real numbers,not all of the same sign,andλ_(1)/λ_(2)is irrational and algebraic.Let V be a well-spaced sequence,δ>0.In this paper,it is proved that,for ...Suppose thatλ_(1),λ_(2),λ_(3),λ_(4),λ_(5)are nonzero real numbers,not all of the same sign,andλ_(1)/λ_(2)is irrational and algebraic.Let V be a well-spaced sequence,δ>0.In this paper,it is proved that,for anyε>0,the number of v∈V with v≤N such that the following inequality|λ_(1)p_(1)~2+λ_(2)p_(2)~2+λ_(3)p_(3)~4+λ_(4)p_(4)~4+λ_5p_5~4-v|<v^(-δ)has no solution in prime variables p_(1),p_(2),p_(3),p_(4),p_(5)does not exceed O(N^(29/32+2δ+ε)).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
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
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
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
基金Supported by the National Natural Science Foundation of China(Grant Nos.1137129161202437)
文摘The main purpose of this paper is using estimates for trigonometric sums and properties of congruence to study the computation of one kind of fourth power mean of a generalized three-term exponential sum, and give an interesting identity for it.
基金Supported by the National Natural Science Foundation of China(Grant No.11571277)the Science and Technology Program of Shaanxi Province(Grant Nos.2014JM1007+2 种基金2014KJXX-612016GY-0802016GY-077)
文摘The computational problem of fourth power mean of generalized three-term exponential sums is studied by using the trigonometric identity and the properties of the reduced residue system. Some explicit formulas for the fourth power mean of generalized three-term exponential sums under different conditions are given.
文摘Let G be a strongly connected directed weighted graph with vertex set{v_(1),v_(2),...,v_(n)},in which each edge e is assigned with an arbitrary nonzero weight w(e).For any two vertices v_i,v_(j )of G,the distance d_(ij )from v_(i)to v_(j)is defined as dij=P∈P(v_(i),v_(j))min∑w(e) where P(v_(i),v_(j)) denotes the set consisting of all the directed paths from v_(i)to v_(j)in G.Given a nonzero indeterminant q,following the definitions from Yan and Yeh (Adv.Appl.Math.,2007),and Bapat et al.(Linear Algebra Appl.,2006),one can define the exponential distance matrix of G as F^(q)_(G)=(q^(dij))_(n×n),and define the q-distance matrix of G as D_(G)^(q)=(d_(ij)^(q))_(n×n)with d_(ij)^(q)={1-q^(dij)/1-q,if q≠1,dij,if q=1,extending the original definitions only for the undirected unweighted connected graphs.One of the remarkable results about the distance matrices of graphs is due to the Graham-HoffmanHosoya theorem (J.Graph Theory,1977).In this paper,we present some Graham-HoffmanHosoya type theorems for the exponential distance matrix F_(G)^(q)and q-distance matrix D_(G)^(q),extending all the known Graham-Hoffman-Hosoya type theorems.
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 ( 11061007)
文摘In this paper,an exponential inequality for weighted sums of identically distributed NOD （negatively orthant dependent） random variables is established,by which we obtain the almost sure convergence rate of which reaches the available one for independent random variables in terms of Berstein type inequality. As application,we obtain the relevant exponential inequality for Priestley-Chao estimator of nonparametric regression estimate under NOD samples,from which the strong consistency rate is also obtained.
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.
基金Supported by NSFC(Nos.12301006,12471009,12071238,11901566,12001047,11971476)Beijing Natural Science Foundation(No.1242003)。
文摘Suppose thatλ_(1),λ_(2),λ_(3),λ_(4),λ_(5)are nonzero real numbers,not all of the same sign,andλ_(1)/λ_(2)is irrational and algebraic.Let V be a well-spaced sequence,δ>0.In this paper,it is proved that,for anyε>0,the number of v∈V with v≤N such that the following inequality|λ_(1)p_(1)~2+λ_(2)p_(2)~2+λ_(3)p_(3)~4+λ_(4)p_(4)~4+λ_5p_5~4-v|<v^(-δ)has no solution in prime variables p_(1),p_(2),p_(3),p_(4),p_(5)does not exceed O(N^(29/32+2δ+ε)).
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.
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.
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.
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.
