算术数列中三个或多个素数的和

Sums of three or more primes in arithmetical progressions

作为圆法的应用,考虑算术数列中的素变数方程p1+p2+…+pk=N,pj≡gj(modh),j=1,2,…,k,∑1≤j≤kgj≡N(modh),k≥3,利用FRIEDLANDER和GOLDSTON的方法给出了方程解数的渐近公式:设k≥3,Θ=sup{β:L(β+iγ)=0},ε>0,h是给定的正整数,则∑p1+p2+…+pk=N,pj≤N,pj≡gj(modh),1≤j≤k(lnp1)(lnp2).….(lnpk)=((k-1)!)-1Nk-1G(k,N)+O(Nk-2+Θ+ε+Nηk+ε),其中G(k,N)是奇异级数,η3=9/5,η4=13/5,ηk=0(k≥5).

As one application of circle method, the object of this paper is to consider the equation (p_1+)p_2+…+p_k=N with p_j≡g_j(mod()h), j=1,2,…,k, ∑_(1≤j≤k)g_j≡N(mod()h), k≥3, and to give a representable asymptotic formula by means of the method of FRIEDLANDER and GOLDSTON. That is, suppose k≥3, Θ=sup{β:L(β+iγ)=0}, ε>0, h is a given positive integer, then ∑_(p_1+p_2+…+p_k=N,p_j≤N,p_j≡g_j(mod h),1≤j≤k)(ln()p_1)(ln()p_2)·…·(ln()p_k)= ((k-1)!)^(-1)N^(k-1)G(k,N)+O(N^(k-2+Θ+ε)+N^(η_k+ε)), where G(k,N) is the singular series, η_3=9/5, η_4=13/5, η_k=0 (k≥5).