Uniform mean value estimates and discrete Hilbert inequalities via orthogonal Dirichlet series (Q397045)

In this paper the author studies series connected with the Dirichlet polynomials of the form \(P(t)=\sum_{n=1}^{m}a_{n}\lambda_{n}^{-it}.\) He proves three theorems and two corollaries; the first of them is Theorem 1.1 which states the following. Let \(r>0\), \(\{a_{n}\}_{n\geq1}\subset\mathbb{C}\), \(\lambda_{0}=0\), \(1=\lambda_{1}<\lambda_{2}<\dots; \lambda_{k}\rightarrow\infty\), and let \(F(t)=\sum_{n=1}^{\infty} a_{n}\lambda_{n}^{-it}.\) Then \[ \int_{-\infty}^{\infty}| F(rt)| ^{2} \frac{dt}{\pi(1+t^{2})}=\sum_{k=1}^{\infty}(\lambda_{k}^{2r}-\lambda_{k-1}^{2r})\big| \sum_{n=k}^{\infty}\frac{a_{n}} {\lambda_{n}^{r}}\big| ^{2} \] if the right-hand side is convergent.