关于一个素数和一个素数的k次方和问题(英文)

On the Sum of a Prime and a κ-power of a Prime

设k≥2,H_k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(modq)在模q的既约剩余系中有解a,b.D_k(N)表示n≤N,n∈H_k,但不能表成p_1+p_2=n的数的个数,其中p_1,p_2表示素数。则在GRH下, D_k(N) N~[1-(1/(h(k)+1))]+ε,这里k=2,3;h(2)=2,h(3)=8.

Let k≥2, H_k denote the set of all numbers n such that a+b^k≡n(modq) has solutions in reduced residues a, b(modq) for any integer q. Let D_k(N) be the number of all n≤N, n∈H_k which cannot be written as p_1+p_2~k=n, where p_1, p_2 denote primes. Then assuming GRH, D_k(N)<< N~[1-(1/k(h(k)+1))]+ε, k=2, 3 for h(2)=2, h(3)=8.