1.Introduction Question: Vinogradov mean value Let $latex k,s\in \mathbb N$,$latex x\in R^k $. $latex J_{s,k}(N)=|\{(n_1,…,n_s,n_{s+1},…,n_{2s})|n_1^j+…+n_s^j=n_{s+1}^j+…+n_{2s}^j) \forall 1\leq j\leq k,1\leq n_i\leq N(1\leq i\leq s) \}|$ How to estimate $latex J_{s,k}(N)$? We assume $latex f_{k}(x,N)=\sum_{1\leq n\leq N}e(nx_1+n^2x_2+…+n^kx_k)$, then by following clear calculate: $latex \int_{[0,1]^k}|f_k(x,N)|^{2s}dx_1…dx_k =\int_{[0,1]^k}|\sum_{1\leq…
