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