 Mean value of Dirichlet polynomials

For finite Dirichlet polynomials, ${\int }_T^{2T}A(\sigma +it)\overline{B(\sigma +it)}dt$ has diagonal term $T{\sum }_n{a}_n\overline{b_n}/n^{2\sigma }$. Its off-diagonal error is bounded by ${\sum }_{m\ne n}|a_n{b}_m|/(n^{\sigma }{m}^{\sigma }|\log (m/n)|)$. A weighted row-sum estimate further bounds this by $\log (2X){\sum }_{n\le X}{n}^{1-2\sigma }(|a_n{|}^2+|b_n{|}^2)$.