Dirichlet polynomial (${\sum }_{n\le X}{a}_n{n}^{-s}$)

A Dirichlet polynomial is a finite Dirichlet series, $A(s)={\sum }_{n\le X}{a}_n{n}^{-s}$. Vertical integration separates equal product indices from oscillatory unequal ones.

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 $\log (2X){\sum }_{n\le X}{n}^{1-2\sigma }(|a_n{|}^2+|b_n{|}^2)$.

For odd $j$, expanding $({\sum }_{p\le X}{p}^{-\sigma }\cos (t\log p){)}^j$ gives products with different numbers of prime factors on the two sides. Unique prime factorization rules out every diagonal equality. The integral over $[T,2T]$ is therefore $O_j(X^j({\sum }_{n\le X^j}{n}^{-\sigma }{)}^2)$.