Let k≥1 be an integer.Assume that RH holds.In this paper we prove that a suitable asymptotic formula for the average number of representations of integers n=p^(k)_(1)+p^(3)_(2)+p^(3)_(3)+p^(3)_(4)+p^(3)_(5),where p_(...Let k≥1 be an integer.Assume that RH holds.In this paper we prove that a suitable asymptotic formula for the average number of representations of integers n=p^(k)_(1)+p^(3)_(2)+p^(3)_(3)+p^(3)_(4)+p^(3)_(5),where p_(1),p_(2),p_(3),p_(4),p_(5)are prime numbers.This expands the previous results.展开更多
In this paper, we give an explicit numerical upper bound for the moduli of arithmetic progressions, in which the ternary Goldbach problem is solvable. Our result implies a quantitative upper bound for the Linnik const...In this paper, we give an explicit numerical upper bound for the moduli of arithmetic progressions, in which the ternary Goldbach problem is solvable. Our result implies a quantitative upper bound for the Linnik constant.展开更多
In this paper, it is proved that with at most O（N65/66） exceptions, all even positive integers up to N are expressible in the form p^2 2＋p^3 3＋p^4 4＋p^5 5. This improves a recent result O（N19193/19200＋ε） due...In this paper, it is proved that with at most O（N65/66） exceptions, all even positive integers up to N are expressible in the form p^2 2＋p^3 3＋p^4 4＋p^5 5. This improves a recent result O（N19193/19200＋ε） due to C. Bauer.展开更多
We consider exceptional sets in the Waring-Goldbach problem for fifth powers.For example,we prove that all but O(N^(131/132))integers satisfying the necessary local conditions can be represented as the sum of 11 fifth...We consider exceptional sets in the Waring-Goldbach problem for fifth powers.For example,we prove that all but O(N^(131/132))integers satisfying the necessary local conditions can be represented as the sum of 11 fifth powers of primes,which improves the previous results due to A.V.Kumchev[Canad.J.Math.,2005,57:298–327]and Z.X.Liu[Int.J.Number Theory,2012,8:1247–1256].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
In this paper it is proved that every sufficiently large even integer N satisfying one of the congruence conditions N ≡ 10, 58, 130, or 178(mod 240) may be represented as the sum of one square and nine fourth powers ...In this paper it is proved that every sufficiently large even integer N satisfying one of the congruence conditions N ≡ 10, 58, 130, or 178(mod 240) may be represented as the sum of one square and nine fourth powers of prime numbers.展开更多
Let Pr denote an almost prime with at most r prime factors, counted according to multiplicity. In the present paper, it is proved that for any sufficiently large even integer n,the equation n = x^3 + p_1~3+ P_2~3+ P_3...Let Pr denote an almost prime with at most r prime factors, counted according to multiplicity. In the present paper, it is proved that for any sufficiently large even integer n,the equation n = x^3 + p_1~3+ P_2~3+ P_3~3 + p_4~3+ P_5~3 + P_6~4 +p_7~4 has solutions in primes pi with x being a P_6. This result constitutes a refinement upon that of Hooley C.展开更多
It is generally known that under the generalized Riemann hypothesis one could establish the twin primes conjecture by the circle method, provided one could obtain the estimate o (nlog-2 n)?for the integral of the repr...It is generally known that under the generalized Riemann hypothesis one could establish the twin primes conjecture by the circle method, provided one could obtain the estimate o (nlog-2 n)?for the integral of the representation function over the minor arcs. One of the new results here is that the assumption of GRH can be removed. We compare this and other such sufficiency results with similar results for the Goldbach conjecture.展开更多
This work presents a fast Cartesian grid-based integral equation method for unbounded interface problems with non-homogeneous source terms.The unbounded interface problem is solved with boundary integral equation meth...This work presents a fast Cartesian grid-based integral equation method for unbounded interface problems with non-homogeneous source terms.The unbounded interface problem is solved with boundary integral equation methods such that infinite boundary conditions are satisfied naturally.This work overcomes two difficulties.The first difficulty is the evaluation of singular integrals.Boundary and volume integrals are transformed into equivalent but much simpler bounded interface problems on rectangular domains,which are solved with FFT-based finite difference solvers.The second one is the expensive computational cost for volume integrals.Despite the use of efficient interface problem solvers,the evaluation for volume integrals is still expensive due to the evaluation of boundary conditions for the simple interface problem.The problem is alleviated by introducing an auxiliary circle as a bridge to indirectly evaluate boundary conditions.Since solving boundary integral equations on a circular boundary is so accurate,one only needs to select a fixed number of points for the discretization of the circle to reduce the computational cost.Numerical examples are presented to demonstrate the efficiency and the second-order accuracy of the proposed numerical method.展开更多
基金the National Natural Science Foundation of China(Grant No.11761048)the Natural Science Foundation of Jiangxi Province for Distinguished Young Scholars(Grant No.20212ACB211007)Natural Science Foundation of Jiangxi Province(Grant No.20224BAB201001).
文摘Let k≥1 be an integer.Assume that RH holds.In this paper we prove that a suitable asymptotic formula for the average number of representations of integers n=p^(k)_(1)+p^(3)_(2)+p^(3)_(3)+p^(3)_(4)+p^(3)_(5),where p_(1),p_(2),p_(3),p_(4),p_(5)are prime numbers.This expands the previous results.
基金Project supported partially by NNSF of China NSF of Henan Province
文摘In this paper, we give an explicit numerical upper bound for the moduli of arithmetic progressions, in which the ternary Goldbach problem is solvable. Our result implies a quantitative upper bound for the Linnik constant.
基金Supported by Post-Doctoral Fellowship of The University of Hong KongThe National Natural Science Foundation(Grant No.10571107)Supported by a grant from the Research Grant Council of Hong Kong(Project No.HKU7028/03P)
文摘In this paper, it is proved that with at most O（N65/66） exceptions, all even positive integers up to N are expressible in the form p^2 2＋p^3 3＋p^4 4＋p^5 5. This improves a recent result O（N19193/19200＋ε） due to C. Bauer.
基金The first author was supported by the Scientific Research Project of the Education Department of Fujian Province(Grant No.JAT190370)the Natural Science Foundation of Fujian Province(Grant No.2020J05162)+1 种基金The second author was supported by the National Natural Science Foundation of China(Grant No.11871367)the Natural Science Foundation of Tianjin City(Grant No.19JCQNJC14200).
文摘We consider exceptional sets in the Waring-Goldbach problem for fifth powers.For example,we prove that all but O(N^(131/132))integers satisfying the necessary local conditions can be represented as the sum of 11 fifth powers of primes,which improves the previous results due to A.V.Kumchev[Canad.J.Math.,2005,57:298–327]and Z.X.Liu[Int.J.Number Theory,2012,8:1247–1256].
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(No.11771333)
文摘In this paper it is proved that every sufficiently large even integer N satisfying one of the congruence conditions N ≡ 10, 58, 130, or 178(mod 240) may be represented as the sum of one square and nine fourth powers of prime numbers.
基金supported by the National Natural Science Foundation of China(No.11201107)the China Scholarship Council
文摘Let Pr denote an almost prime with at most r prime factors, counted according to multiplicity. In the present paper, it is proved that for any sufficiently large even integer n,the equation n = x^3 + p_1~3+ P_2~3+ P_3~3 + p_4~3+ P_5~3 + P_6~4 +p_7~4 has solutions in primes pi with x being a P_6. This result constitutes a refinement upon that of Hooley C.
文摘It is generally known that under the generalized Riemann hypothesis one could establish the twin primes conjecture by the circle method, provided one could obtain the estimate o (nlog-2 n)?for the integral of the representation function over the minor arcs. One of the new results here is that the assumption of GRH can be removed. We compare this and other such sufficiency results with similar results for the Goldbach conjecture.
基金supported by the National Key R&D Program of China(Project No.2020YFA0712000)supported by the Shanghai Science and Technology Innovation Action Plan in Basic Research Area(Project No.22JC1401700)+1 种基金the Strategic Priority Research Program of Chinese Academy of Sciences(Grant No.XDA25010405)the National Natural Science Foundation of China(Grant No.DMS-11771290).
文摘This work presents a fast Cartesian grid-based integral equation method for unbounded interface problems with non-homogeneous source terms.The unbounded interface problem is solved with boundary integral equation methods such that infinite boundary conditions are satisfied naturally.This work overcomes two difficulties.The first difficulty is the evaluation of singular integrals.Boundary and volume integrals are transformed into equivalent but much simpler bounded interface problems on rectangular domains,which are solved with FFT-based finite difference solvers.The second one is the expensive computational cost for volume integrals.Despite the use of efficient interface problem solvers,the evaluation for volume integrals is still expensive due to the evaluation of boundary conditions for the simple interface problem.The problem is alleviated by introducing an auxiliary circle as a bridge to indirectly evaluate boundary conditions.Since solving boundary integral equations on a circular boundary is so accurate,one only needs to select a fixed number of points for the discretization of the circle to reduce the computational cost.Numerical examples are presented to demonstrate the efficiency and the second-order accuracy of the proposed numerical method.
