# Mean squares of certain real character sums

Peng Gao

## Affiliation:

Beihang University

## Date:

Tue, 24/09/2019 - 12:00pm to 1:00pm

## Venue:

RC-4082, The Red Centre, UNSW

## Abstract:

Let $\mathcal{D}$ be the set of non-square quadratic discriminants and $\chi_D=(\frac {D}{\cdot})$ be the Kronecker symbol. The mean square estimation
\begin{align*}
\sum_{\substack {|D| \leq X \\ D \in \mathcal{D}}} \left| \sum_{n \leq Y} \Big (\frac {D}{n} \Big ) \right|^2 \ll XY \log X
\end{align*}
is due to M. V. Armon. The result can be applied to study the mean values of class numbers of imaginary quadratic number fields and the second moment of Dirichlet $L$-functions with primitive quadratic characters. Note that it is relatively easy to obtain an asymptotic formula for the sums considered above when $Y$ is small compared to $X$. In this talk, we show how to obtain an asymptotic formula for such sums when $Y$ is close to $X$.