Almost Equal Linear Goldbach Problem

In 1937,Vinogradov proved that any sufficiently large positive odd integer can be expressed as the sum of three prime numbers,i.e.n=p1+p2+p3.Later,some mathematicians have studied related problems such as restricting pj in short intervals,which is know as almost equal Goldbach problem.Precisely,solvability of the equation n=p1+p2+p3,with|pj-n/3|≤y,and y=o(n).In 1990,Pan and Pan proved that the equation has solution when y(?)n2/3+ε.In 1998,by using sieve methods Baker and Harman showd that y(?)n4/7+ε is sufficient.The best result so far is Maynard’s y=n11/20+ε,In 2005,Meng proved that y(?)n1/2+εis sufficient under the GRH.In 1989,Liu and Tsang investigated the ternary Goldbach problem with coefficients,i.e.a1p1+ a2p2+a3p3=n,aj∈N+.This problem is firstly raised by Baker.In 2005,Liu and Tsang considered this problem with aj not all positive.Motivated by these problems,in this paper we will consider the ternary Goldbach problem with coefficients and almost equal variables,that is,the solvability of equation a1p1+a2p2+a3p3=n,|ajpj-n3|≤y.By using the technique to enlarge the major arcs in circle method and combining some results in almost equal Goldbach problem,we will prove that for(?),the equation is solvable.