Character analogues of certain Hardy-Berndt sums

TL;DR

This paper develops transformation formulas and reciprocity theorems for Hardy-Berndt sums involving characters, providing integral representations and exploring their properties.

Contribution

It introduces character analogues of Hardy-Berndt sums, derives new transformation and reciprocity formulas, and establishes integral representations for these sums.

Findings

Derived transformation formulas for character sums.

Established reciprocity theorems for these sums.

Provided integral representations using Euler--Maclaurin formula.

Abstract

In this paper we consider transformation formulas for \[ B\left( z,s:\chi\right) =\sum\limits_{m=1}^{\infty}\sum\limits_{n=0} ^{\infty}\chi(m)\chi(2n+1)\left( 2n+1\right) ^{s-1}e^{\pi im(2n+1)z/k}. \] We derive reciprocity theorems for the sums arising in these transformation formulas and investigate certain properties of them. With the help of the character analogues of the Euler--Maclaurin summation formula we establish integral representations for the Hardy-Berndt character sums $s_{3,p}\left( d,c:\chi\right) $ and $s_{4,p}\left( d,c:\chi\right) $.