黎曼猜想下一个素数k次幂及四个素数3次幂的华林-哥德巴赫问题

令k≥1是一个整数.在黎曼猜想成立的条件下,本文给出了n=p₁^k+p₂³+p₃³+p₃⁴+p₅³平均表法个数的一个合适的渐近公式,其中p₁,p₂,p₃,p₄,p₅都是素数,推广了之前的结果.     

一个素数和一个素数的平方和问题

H表示一个正整数N的集合,使对任意的正整数q,同余方程a+b~2≡N(mod q)在模q的既约剩余系中有解a;b.E(x)表示N≤x,N∈H,但不能表成p_1+p_2~2=N的数的个数,其中p_1,p_2个表示素数,则E(x)<相似文献   

华林-哥德巴赫问题:两个平方,两个立方和两个四次方

令P_r表示素因子不超过r的殆素数,按重数计.作者证明了对于充分大的偶数N,方程N=x~2+p_1~2+p_2~3+p_3~3+p_4~4+p_5~4有解,其中x是殆素数P_6,p_j(j=1,…,5)是素数.     

一个素数和两个素数的平方和问题

本文证明了最多有O(N13/30+ε)个例外之外,所有的正的奇整数n≤N,n≡0或1(mod 3)能表示成一个素数和两个素数的平方和.     

几乎相等的三次华林-哥德巴赫问题的例外集

本文证明了对5≤s≤8,几乎所有的满足某些同余条件的正整数N都可以表示为N=p31+···+p3s,|pi-(N/s)1/3|≤N1/3-θs,其中θ5=7261-2ε,θ6=5159-ε,θ7=11333-ε,θ8=19561-ε.     

华林问题中的一些新结果

设Ｇ（ｋ）为所有充分大的正整数均可表示为不超过ｓ个正整数ｋ次方幂之和的最小ｓ．本文对Ｇ（１２），Ｇ（１３），Ｇ（１９）给出了新的估计．     

华林—哥德巴赫问题:1个立方与9个四次方

设N为充分大的整数.本文证明了对于不超过N且满足某些必要同余条件的正整数n除至多O(N^17/18+ε)之外均可以表示为p₁³+p₂⁴+p₃⁴+p₄⁴+p₅⁴+p₆⁴+p₇⁴+p₈⁴+p₉⁴+p₁₀⁴的形式,其中p₁,p₂,…,p₁₀为素数.     

关于一个素数和一个素数的k次方和问题

设k≥2,且Hk表示一个正整数n的集合,使得该集合中的元素满足a+bk≡n(modq)对任意的q,在模q的既约剩余系中有解,令Dk(N)表示所有的n≤N,且n∈Hk且不能表成p1+p2k=n形式的整数.那么在GRH下, Dk(N)相似文献   

小区间上的素变数三角和估计

设A(n)为von Mangoldt函数且实数θ=95-83~(1/2)/121.当xθ+ε≤y≤x时,本文对于所有的α∈[0,1]给出了指数和S2(x,y;α)=∑x0,估计式∑x相似文献   

关于一个素数和一个素数的k次方和问题

设k≥2,Hk表示一个正整数n的集合,使对任意的正整数q,同余方程a+b2三n(modq)在模q的既约剩余系中有解a,b.Dk(N)表示n≤N,n∈Hk,但不能表成p1+p22=n的数的个数,其中p1,p2表示素数.则在GRH下,Dk(N)<<N1-1/k(h(k)+1)+ε,这里k=2,3;h(2)=2,h(3)=8.     

On sums of a prime and four prime squares in short intervals

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.     

Exponential sums over primes in short intervals

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.     

On sums of a prime and four prime squares in short intervals

In this paper, we are able to sharpen Hua's classical result by showing that each sufficiently large integer can be written as

     

小区间上的三素数定理

设N是充分大的奇数,本文在广义Riemman 假设下证明了方程N=p1 p2 p3,pi-(N/3)≤U, i=1,2,3, U≥N(1/2)logN3 ε有解,此处pi是素数,并得到了方程解数的渐进式.     

On exponential sums over primes in short intervals

Let Λ(n) be the von Mangoldt function, x real and y small compared with x. This paper gives a non-trivial estimate on the exponential sum over primes in short intervals for all α ∈ [0,1] whenever . This result is as good as what was previously derived from the Generalized Riemann Hypothesis.     

On exponential sums over primes in short intervals

Let Λ(n) be the von Mangoldt function, x real and y small compared with x. This paper gives a non-trivial estimate on the exponential sum over primes in short intervals S₂(x,y;a)=?_{x < n ￡ x+y}L(n)e(n² a)S_2(x,y;{\alpha})=\sum_{x < n \le x+y}\Lambda(n)e(n^2 {\alpha}) for all α ∈ [0,1] whenever x^\frac23+e ￡ y ￡ xx^{\frac{2}{3}+{\varepsilon}}\le y \le x . This result is as good as what was previously derived from the Generalized Riemann Hypothesis.     

On Diophantine approximation with one prime and three squares of primes

Let λ₁, λ₂, λ₃, λ₄ be non-zero real numbers, not all of the same sign, w real. Suppose that the ratios λ₁/λ₂, λ₁/λ₃ are irrational and algebraic. Then there are in.nitely many solutions in primes p_j, j =1, 2, 3, 4, to the inequality . This improves the earlier result.     

Sums of primes and squares of primes in short intervals

Let H₂{\mathcal H_2} denote the set of even integers n \not o 1 mod 3{n \not\equiv 1 \pmod 3} . We prove that when H ≥ X ^0.33, almost all integers n ? H₂ ?(X, X + H]{n \in \mathcal H_2 \cap (X, X + H]} can be represented as the sum of a prime and the square of a prime. We also prove a similar result for sums of three squares of primes.