A transition formula for mean values of Dirichlet polynomials

Let \[ f(t)=\sum_{N n 2N}a_nn^{-it} \] be a Dirichlet polynomial. We consider the weighted square mean value \[ I=\int_{-\infty}^{\infty}|f(t)|^2\exp\{-\Delta^{-2}(t-T)^2\}\,dt, \] where \(T\) is a large paremeter and \[ \Delta = \frac{T}{\log T}. \] Assume \[ N=T^c \] with a constant \(c > 0\). In the case \(c 1\), an asymptotic formula for \(I\) can be obtained via classical methods. On the other hand, for \(c > 1\) we have \[ I\ll N\sum_{N n 2N}|a_n|^2. \] This bound can not be substantially sharpened in general. The aim of this talk is to introduce a formula that transfers the upper bound for \(I\) (with \(c > 1\)) to a square mean of the linear exponential sum \[ \sum_{N n 2N}a_ne(-\alpha n), \] where \(e(x)=\exp\{2\pi ix\}\).