Character Average of Second and Fourth Powers of Dirichlet L-Series at Unity

TL;DR

This paper derives new elementary formulas for the average values of the second and fourth powers of Dirichlet L-series at unity, providing insights into their behavior through novel methods.

Contribution

It introduces simple elementary techniques to compute character averages of Dirichlet L-series powers at unity, including a new result for the fourth power.

Findings

Character average of the fourth power of Dirichlet L-series at unity obtained.

Character average of the second power of Dirichlet L-series at unity derived using Hurwitz zeta function.

Results appear to be new contributions to the understanding of Dirichlet L-series.

Abstract

We obtain by simple elementary method, the character average of fourth power of Dirichlet L-series at unity.This result seems to be new.We also obtain character average of second power of Dirichlet L-series at unity,using the power series in second variable of Hurwitz zeta function at unity.