Mean values of some Hecke characters

TL;DR

This paper evaluates a smoothed sum involving quadratic, cubic, or quartic Hecke characters, providing insights into their mean values when summing over two comparable ranges.

Contribution

It introduces a new evaluation method for smoothed character sums involving Hecke characters of various orders, extending previous work to more general characters.

Findings

Derived explicit formulas for smoothed sums of Hecke characters.

Extended understanding of mean values of Hecke characters.

Provided techniques applicable to related character sum problems.

Abstract

In this paper, we evaluate a smoothed character sum involving $\sum_{m}\sum_{n}\leg {m}{n}$, with quadratic, cubic or quartic Hecke characters $\leg {m}{n}$, and the two sums over $m$ and $n$ are of comparable lengths.