【摘要】 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).更多还原

【Abstract】 As one application of circle method, the object of this paper is to consider the equation （p₁+）p₂+…+p_k=N with p_j≡g_j（mod（）h）, j=1,2,…,k, ∑_1≤j≤kg_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 ∑_{p1+p2+…+pk=N,pj≤N,pj≡gj（mod h）,1≤j≤k}（ln（）p₁）（ln（）p₂）·…·（ln（）p_k）= （（k-1）!）^-1N^k-1G（k,N）+O(N^k-2+Θ+ε+N^η_k+ε), where G（k,N） is the singular series, η₃=9/5, η₄=13/5, η_k=0 （k≥5）.更多还原