# A transition formula for mean values of Dirichlet polynomials

 Joint IAS/Princeton University Number Theory Seminar Topic: A transition formula for mean values of Dirichlet polynomials Speaker: Yitang Zhang Affiliation: University of New Hampshire Date: Monday, April 21 Time/Room: 3:30pm - 4:30pm/S-101 Video Link: http://video.ias.edu/jointiasnts/2014/0421-TomZhang

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 $e(x)=\exp\{2\pi ix\}$.