Denote by M the 'second main term' attached to the Siegel zeros of Dirichlet L-functions in odd Goldbach problem. Let N≥exp(exp(9.873)) be a real number, r=logN, the statement M≤0.032 N^2r^(-3) is proved in ...Denote by M the 'second main term' attached to the Siegel zeros of Dirichlet L-functions in odd Goldbach problem. Let N≥exp(exp(9.873)) be a real number, r=logN, the statement M≤0.032 N^2r^(-3) is proved in this paper.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
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, 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
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 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
In this paper, we consider the Goldbach＇s problem for matrix rings, namely, we decompose an n × n （n 〉 1） matrix over a principal ideal domain R into a sum of two matrices in Mn,（R） with given determinants....In this paper, we consider the Goldbach＇s problem for matrix rings, namely, we decompose an n × n （n 〉 1） matrix over a principal ideal domain R into a sum of two matrices in Mn,（R） with given determinants. We prove the following result： Let n 〉 1 be a natural number and A = （aij） be a matrix in Mn（R）. Define d（A） ：= g.c.d{aij}. Suppose that p and q are two elements in R. Then （1） If n 〉 1 is even, then A can be written as a sum of two matrices X, Y in Mn（R） with det（X） = p and det（Y） = q if and only if d（A） [ p - q; （2） If n 〉 1 is odd, then A can be written as a sum of two matrices X, Y in Mn（R） with det（X） = p and det（Y） = q if and only if d（A） ｜ p ＋ q. We apply the result to the matrices in Mn（Z） and Mn（Q[x]） and prove that if R = 7. or Q[x], then any nonzero matrix A in Mn（R） can be written as a sum of two matrices in Mn（R） with prime determinants.展开更多
文摘Denote by M the 'second main term' attached to the Siegel zeros of Dirichlet L-functions in odd Goldbach problem. Let N≥exp(exp(9.873)) be a real number, r=logN, the statement M≤0.032 N^2r^(-3) is proved in this paper.
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 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(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.
文摘In this paper, we consider the Goldbach＇s problem for matrix rings, namely, we decompose an n × n （n 〉 1） matrix over a principal ideal domain R into a sum of two matrices in Mn,（R） with given determinants. We prove the following result： Let n 〉 1 be a natural number and A = （aij） be a matrix in Mn（R）. Define d（A） ：= g.c.d{aij}. Suppose that p and q are two elements in R. Then （1） If n 〉 1 is even, then A can be written as a sum of two matrices X, Y in Mn（R） with det（X） = p and det（Y） = q if and only if d（A） [ p - q; （2） If n 〉 1 is odd, then A can be written as a sum of two matrices X, Y in Mn（R） with det（X） = p and det（Y） = q if and only if d（A） ｜ p ＋ q. We apply the result to the matrices in Mn（Z） and Mn（Q[x]） and prove that if R = 7. or Q[x], then any nonzero matrix A in Mn（R） can be written as a sum of two matrices in Mn（R） with prime determinants.
基金Supported by NSFC(No.11201107.No.11271283)the Natural Science Basic Research Plan in Shaanxi Province of China(No.2016JM1013)+1 种基金the Starting Research Fund for Doctors of Xi'an Polytechnic University(No.BS1508)the Quality Improvement Project for Basic Curriculum(Higher Mathematics)of Xi'an Polytechnic University
