Lethbridge Number Theory and Combinatorics Seminar: Nathan Ng

A Dirichlet polynomial is a function of the form $A(t)=\sum_{n \le N} a_n n^{-it}$ where $a_n$ is a complex sequence, $N \in \mathbb{N}$, and $t \in \mathbb{R}$. 

For $T \ge 1$, the mean values

$$\int_{0}^{T} |A(t)|^2 \, dt$$

play an important role in the theory of L-functions.  I will discuss work of Goldston and Gonek on how to evaluate these integrals in the case that $T < N < T^2$.  This will depend on the correlation sums \[

   \sum_{n \le x} a_n a_{n+h} \text{ for } h \in \mathbb{N}.

\]

If time permits, I will discuss a conjecture of Conrey and Keating in the case that $a_n$ corresponds to a generalized divisor function and $N > T$.