Mean value of Dirichlet polynomials
= 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 $\log(2X)\sum_{n\le X}n^{1-2\sigma}(|a_n|^2+|b_n|^2)$.